Progress in nonlinear analysis : proceedings of the Second International Conference on Nonlinear Analysis, Tianjin, China, 14-19 June 1999

PROGRESS IN NONLINEAR ANALYSIS

NANKAI SERIES IN PURE, APPLIED MATHEMATICS AND THEORETICAL PHYSICS Editors: S. S. Chern, C. N. Yang, M. L. Ge, Y. Long Published:

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Probability and Statistics eds. Z. P. Jiang, S. J. Yan, P. Cheng and R. Wu

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Nonlinear Analysis and Microlocal Analysis eds. K. C. Chung, Y. M. Huang and T. T. Li

Vol. 3:

Algebraic Geometry and Algebraic Number Theory eds. K. Q. Feng and K. Z. Li

Vol. 4:

Dynamical Systems eds. S. T. Liao, Y. Q. Ye and T. R. Ding

Vol. 5:

Computer Mathematics eds. w.-T. Wu and G.-D. Hu

Vol. 6:

Progress in Nonlinear Analysis eds. K. -c. Chang and Y. Long

Proceedings of the Second International Conference on Nonlinear Analysis lianjin, China

14-19 June 1999

PROGRESS IN NONLINEAR ANALYSIS Edited by

Kung-Ching Chang Peking University, China

Yiming Long Nankai University, China

\\1» World Scientific ••

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Nankai Series in Pure, Applied Mathematics and Theoretical Physics - Vol. 6 PROGRESS IN NONLINEAR ANALYSIS Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd.

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Professor Paul H. Rabinowitz

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PREFACE The Second International Conference on Nonlinear Analysis (SICNAT) was held from June 14th to 19th of 1999 at Nankai University, Tianjin, China. Forty one invited speakers from twelve countries and areas in the world gave their lectures in the conference. This conference mainly concentrated on the following areas in nonlinear analysis: critical point theory, Hamiltonian systems, KAM theory, elliptic equations and systems, geometrical analysis, and global bifurcation theory. Before the conference, a two week special workshop was held from May 31st to June 11th of 1999 at Nankai to give basic short courses on critical point theory, homo clinic orbits, KAM theory, and celestial mechanics. More than thirty young scholars and graduate students from different universities and institutes in China joined the workshop and the conference. The lectures presented in the conference and minicourses given in the workshop reflect from different angles the broad and rapid development of nonlinear analysis. This volume contains most of these lectures. During the conference, a special party was held for Professor Paul H. Rabinowitz on the occasion of his sixtieth birth day, to whom this book is dedicated. HIS outstanding contributions in the last thirty years has largely influenced the nonlinear analysis. The conference and the workshop were sponsored by: K. C. Wong Education Foundation

National Natural Science Foundation of China Mathematical Center of Ministry of Education of China Tianjin City Education Committee Nankai University Nankai Institute of Mathematics. We wish to thank all these institutions for their generous financial support which made the conference possible. We extend our thanks to all speakers of the conference and the workshop, all the related personnel in Nankai who greatly contributed to the success of the conference and the workshop, and World Scientific, specially the editor Ms. E. H. Chionh, who greatly helped us on the publication of this proceedings volume. Kung-Ching Chang February 2000

Yiming Long vii

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CONTENTS Preface

vii

Extremal Sign Changing Solutions of Semilinear Dirichlet Problems Thomas Bartsch On the Principal Eigenvalue of Elliptic System with Indefinite Weight and Applications Chang K ung- Ching

1

11

The Ricci Flow on Complete Three-Manifolds Bing-Long Chen and Xi-Ping Zhu

24

Heteroclinic Orbits of Second Order Hamiltonian Systems Chao-Nien Chen

46

On the Connecting Orbits Among Local Minimizers for Monotone Twist Map Wei Cheng and Chong-Qing Cheng

58

Multiple Solutions and Bifurcation of Nonhomogeneous Semilinear Elliptic Equations in ]RN Yinbin Deng, Yi Li and X uejin Zhao

81

Properties of Solutions of Some Forced Nonlinear Oscillators at Resonance C. Fabry and J. Mawhin Variational Sl-symmetric Problems with Resonance at Infinity Norimichi Hirano and Slawomir Rybicki Multiplicity Results for Hamiltonian Systems with Nonconvex Hamiltonian Functions Marek Izydorek

103

119

149

The Dynamics of the Flow for Prescribed Harmonic Mean Curvature Huai- Yu Jian and Bin-Heng Song

166

A Symplectic Transformation and Its Applications Mei- Yue Jiang

175

ix

x Positive Solution to p-Laplacian Type Scalar Field Equation in RN with Nonlinearity Asymptotic to u p - 1 at Infinity Gongbao Li, Lina Wu and Huan-Song Zhou

Contents

188

The Existence and Convergence of Heat Flows Jiayu Li

215

Some Aspects of Semilinear Elliptic Boundary Value Problem Shujie Li

234

Iteration Formula for the w Index with Applications Chun-Gen Liu

257

Dynamics on Compact Convex Hypersurfaces in R 2n Yiming Long

271

Gromov-Witten Invariants on Compact Symplectic Manifolds with Contact Type Boundaries and Applications Guangcun Lu Exact Lagrangian Sub manifolds in Symplectizations Renyi Ma Low Dimension Anomalies and Solvability in Higher Dimensions for Some Perturbed Pohozaev Equation Gianni Mancini

289

303

315

A New Proof of a Theorem of Strobel Paul Rabinowitz

326

Global Injectivity and Asymptotic Stability via Minimax Method Elves A . B. Silva and Marco A. Teixeira

339

Prescribed Energy Problem for a Singular Hamiltonian System of 2-body Type }(azunaga Tanaka

359

Sign-Changing Solutions for a Class of Nonlinear Elliptic Problems Zhi-Qiang Wang

370

A Class of Resonant or Indefinite Elliptic Problems Shaoping Wu

384

Contents

xi

Geometric Structures of Solutions for Certain Elliptic Differential Equations Xue-Feng Yang

394

KAM Theory for Lower Dimensional Tori of Nearly Integrable Hamiltonian Systems Jiangong You

409

Periodic Solutions for N-body Problems Shiqing Zhang

423

List of Scientific and Organizing Committees

445

List of Speakers

445

Extremal sign changing solutions of semilinear Dirichlet problems Thomas Bartsch Mathematisches Institut Universitat Giessen Arndtstr. 2 35392 Giessen Germany

1

Introduction

We consider the Dirichlet problem

= f(x, u) u=o

-~u

{

in 0 on

ao

(D)

where 0 c IIlN is a bounded domain with smooth boundary and f : 0 x III ~ III is continuous. In recent years several papers appeared in which variational methods are combined with sub-/supersolution techniques. The goal of these papers is to obtain additional information on the solutions obtained via variational methods, for example, that a solution changes sign, or that it lies outside of a given order interval. This information can be used to obtain new multiplicity results. The idea goes back to Hofer [20] and Cerami-Solimini-Struwe [12]; see [3]-[6], [8]-[10], [13]-[19], [23] for recent work. In this note we are interested in a special type of sign changing solution. Let SC- (respectively SC+) denote the set of subsolutions (respectively supersolutions) of (D) which change sign. We call a sign changing solution u of (D) extremal if it is a maximal element of SC+ and a minimal element of SC- . Thus if l!c is a subsolution and l!c < u then l!c < O. Similarly, if u > u is a supersolution then u > O. Consequently, u is both maximal and minimal in the set SC = SC+ n SC- of all sign changing solutions of (D) . It has been proved in [3] and [6] that certain min-max procedures can be refined in order to yield 1

Thomas Bartsch

2

an extremal sign changing solution. In this note we present a new approach to the existence of extremal sign changing solutions. This will be used to present simple proofs of two results from [3] and [6].

2

The almost connecting orbits method

Let X be a Banach space and P C X a closed cone, that is, P = P is convex, R+ . PCP and P n (-P) = {a} . Suppose P has non empty interior int(P) . We use the notation Do := P u (-P) and Du := u + Do. If v E Du then u and v are said to be comparable with respect to the partial order u

~

v

:¢:::::>

v- u EP

We also write u « v if v - u E int(P). Elements of P := P" {a} are called positive, elements of - P negative, and elements of X " Do sign changing. A map f : X -+ X is said to be order preserving if u ~ v implies f (u) ~ f (v) . It is called strongly order preserving if u < v implies feu) « f(v). We refer the reader to the survey [1] by Amann on order structures, fixed point theory of order preserving maps and applications to (D). We also refer to the classical work of Krasnoselski [21] and of Krein-Rutman [22] for the foundations of the subject. Now let E be a Hilbert space with X C E being densely embedded. We consider a functional : E -+ IR satisfying

(d E CI ,I(E) and its gradient has the form V' = 1- K where K(X) eX and the restriction Klx : X -+ X is strongly order preserving. A point u E X is said to be sub critical if V'(u) ~ O. It is called supercritical if V'(u) ~ O. We write SC- (respectively SC+) for the set of sign changing sub critical (respectively supercritical) points of . Let cpt(u) be the flow on E defined by

{!

cpt (u) = -x(IIV'( cpt (u)) IIE)V'( cpt (u))

cpo(u)

=u

where X : IR -+ IR is given by x(r) = 1 for r ~ 1 and x(r) = 1/r for r The next condition is a linking condition.

> 1.

(2) There exists Uo EX, a compact set C C X and a set SeE such that

(i) cpt(u) -+ Uo as t -+

00

for all u E Sj

Extremal sign changing solutions of semilinear Dirichlet problems (ii) inf 9 Is

> 9(uo); (iii) cpt(C) n S # 0 for all t 2 0. As a consequence of (az)(i) the point uo is a critical point of 9 . The set S is usually a sphere in the stable manifold of uo. In our applications the set C is the convex hull of a set B which links with S. The sets B and S need not be separated by levels of 9 , that is, the inequality max 91 < inf 91 need not hold. It follows from (92)(iii) and the compactness of C that there exist sequences (v,), in C and (t,), in R such that cptn(vn) E S for all n and v, + v E C , t, + oo as n + m. This implies 9(cpt(v)) > i n f a i s = : a for a l l t 2 0

(2.1)

because if 9((pT (v)) < a for some T 2 0 then 9(cpT(u)) < a provided Ilu vll < E is small. Therefore 9(cpT(vn)) < a if n is large enough which implies 9(cptn(vn)) < a if n is such that 1111, -v1) < E and t, > T. However, cptn (v,) E S and a = inf @Is, a contradiction. From 2.1 it follows that c := limt+a, 9(cpt(v)) 2 a and that there exists a sequence s, + m with

Setting un := cpSn (v) we thus produced a (PS),-sequence (u,),. Now suppose 9 satisfies the Palais-Smale condition (PS),. Then (u,), has a convergent subsequence hence 9 has a critical point u1 on the level c. More precisely, we have a point v E C whose w-limit set w(v) C {u E X : 9(u) = C, V9(u) = 0) is not empty. In addition, v is the limit of a sequence of points (v,), in the stable manifold of UO. We shall now use this sequence of orbits (through v,, n E N) which almost connect u1 to 0 in order to prove a certain extremality property of 211. If u E X is a subcritical point of 9 then it is well known and not difficult to see that .u P is positive invariant under cpt. More precisely, we have the implication

+

Similarly, if 'ii E X is a supercritical point of 9 then 'ii- P is positive invariant in the same strong sense. A simple consequence of this positive invariance is that ul is not comparable to uo provided S n int(D,,) = 0. In order to see this suppose to the contrary

Thomas Bartsch

4

that Ul E Duo, hence Ul E int(Duo) by (2.2). Then 0 with 0 is from (h) . By our hypotheses eI> E C2 (E) and the Palais-Smale condition (PS) c holds for all c E lit Let 'Vel> be the gradient vector field of eI> associated to ( . , . ) E and let cpt (u) be the flow on E defined by

{

!

cpt(u)

= -x(II'VeI>(cpt(u))IIE)'VeI>(cpt(u))

cpO(u)

=u

(3.3)

where X : lR -+ lR is given as in section 2: x(r) = 1 for r ~ 1 and x(r) = 1/r for r > 1. Since the cone PE = {u E E : u 2: 0 a. e.} has empty interior it is convenient to work on X := CJ(IT) = {u E C1 (IT) : ulan = O}. Here the positive cone P = {u EX: u 2: O} = X n PE has non-empty interior. From our regularity hypotheses on f it follows that 'Vel> = 1- K leaves X invariant, hence cpt(X) C X . Moreover, by (h) and our choice of ( ., ·)E we have that Klx is strongly order preserving. Thus (eI>d holds. Now we produce the three critical points with the help of Theorem 2.1 using different choices for the sets C and S appearing in (eI>2). First observe that 0 is a nondegenerate local minimum of eI> since -~ + ft(x, 0) is positive definit. There exists p > 0 such that

cpt(u) -+ 0 as t -+

00

for every u E E

with Ilull

=p

and inf el>ISpE > eI>(O)

=0

where SpE = {u E E : lIull = pl . Moreover, there exists v+ E P C X with Ilv+IIE > p and eI>(v+) ~ O. Now we set B+ = {O,v+}, C+ := conv B+ = {tv+ : o ~ t ~ 1} and S+ := {u E P : IluliE = pl. Clearly, cpt(C+) n S+ :f. 0 for all t ~ 0 because cpt(O) = 0 and cpt(v+) lie in different components of p" S+. Theorem 2.1 yields a critical point u+ which lies in int(P) since C+ C P, and it satisfies property a) by 2.1(iv). The negative solution u_ is obtained analogously. For the sign changing solution Ul we choose Vo E P C X and Vl EX" lRvo with Ilvoll, IIvdl > p and such that

eI>(vo cosw + VI sinw)

~ 0

for all wE [0,11").

Then we set

B1

:

= {vo cos w + VI sin w : 0 ~ w ~ 11"} U {rvo : Ir I ~

and C 1 := conv B 1 . We also set SI := {u E E : u ..i el,

Ilull = p} eX" Do

1}

E x t r e m p e r 1 1 ~ 1 1 1 1 e uirichlet w problems

7

where el E int(P) is the positive eigenvector of -A+ f t ( z , 0 ) in H i (0).Observe that cpt(B1)n S1 = 0 for all t 3 0 because for u = vo cos w vl sin w we have b ( u ) 5 0 , so cpt(u) E b0 c X \ S1 for all t 2 0. And for u = TWO E B1 we have u E DO,hence cpt(u) E Do c X \ S1 for every t 2 0. A simple argument using the Brouwer degree yields cpt(Cl)n S1 # 0 for t 2 0. Theorem 2.1 yields an extremal sign changing solution. The next result is not a direct consequence of Theorem 2.1. Instead we use the almost connecting orbits method in a somewhat more complicated situation.

+

Theorem 3.2 Suppose ( f l ) - ( f 3 ) . I f f is odd in the u-variable then (D) has an unbounded sequence of extremal sign changing solutions.

The existence of the solutions is a well known consequence of the symmetric mountain pass theorem from [2],[24]. That the solutions are extremal sign changing solutions has been proved in [3]using equivariant cohomology theory. There we also obtained information on the critical groups. The proof we give below is more elementary and quite instructive. It does not yield the information on the critical groups however. proof. Let X1 < X z 5 X 3 5 . . . be the eigenvalues of - A f t ( x , 0 ) with eigenfunctions el E P, ez,es,. . . . If X 1 > 0 we set ko := 1, otherwise ko E N is defined by Xko 5 0 < X k o + l . We set E - := spaniel,. . . ,ek,} C X and E+ := ( E - )=lspan{ek : k > ko}. As in the proof of Theorem 3.1 we let V 9 be the gradient vector field associated to . ) E and write cpt(u) for the flow on E defined by (3.1). Next we consider the stable manifold

+

( a ,

W S t := { u E E : eXko+1t/2cpt(~) + 0 in E a s t

+ 00).

According to the stable manifold theorem of Bates and Jones [7] the local stable manifold is described as a graph as follows. There exists p > 0 and a C1-map

such that

We set

As in [3],Lemma 4.3, one proves that S1 n Do = 0. Now let a := inf 9 ( S 1 )> 0 and fix any p > a. We want to prove that 9 has an extremal sign changing

Thomas Bartsch

8

critical point above the level {3. In order to see this we consider the maximal invariant set of cp between the levels a and {3:

1:= {u EX : lJ>(cpt(u)) E [a,{3]

for all t E 1R}.

A standard deformation lemma argument shows that for each c E [a, (3] there exists to > 0 such that

where genus(A) denotes the classical genus of symmetric subsets A = -A of X introduced by Krasnoselski. As a consequence we obtain that 'Y := genus(I) < 00. Define Y := span{ek : 1 :::; k :::; ko From (h) it follows that there exists R

lJ>(u) < inflJ>(BpX)

+ 'Y + I} C

X

> 0 such that

for u E Y

with

lIull

~

R.

(3.4)

Now we consider the set

M:= {u E BRY: lim lJ>(cpt(u)) E [a,(3)} t-+oo

The Palais-Smale-condition implies that for each u E M there exists c .limHoo lJ>(cpt(u)) E [a,{3]' so that cpt(u) -t Kc as t -t 00. From this we easily deduce that for every neighborhood U of 1 there exist T = T(U) > 0 such that cpT (M) C U. Consequently, genus(M) :::; genus(I) = 'Y by the continuity and monotoniciy of the genus. Since M is compact there exists to > 0 with genus(U.(M)) = genus(M) :::; 'Y. Now we claim that for every T > 0 there exists UT E BRY " U.(M) such that cpT(UT) E SI . For this it is enough to show that genus ( {u E BRY : cpt(u) E Sd) ~ 'Y + 1

(3.5)

which implies that {u E BRY : cpT(u) E Sd ¢. U.(M). In order to prove (3.3) we consider the set

OT is an open neighborhood of 0 in Y with OT C intBRY by (3.2). The map

Extremaa

sIgn UldlIglllg SWUC101l5 ill

SCHllJl.UCa.L

.LIlllLhlet problems

9

is continuous and odd; P+ : E ~ E+ is the orthogonal projection. The properties of the genus and Borsuk's theorem imply that

genus(h-1(0)) ~ dim Y - dim E-

= 'Y + l. h(u) = 0 implies cpT(u)

Observe that u E BOT implies IIcpT(u) II = p, and E wst, hence h- 1 (0) C {u E BRY : cpT(u) E St}. This proves (3.3). Now we continue as in section 2. There exists a sequence Vn E BRY" Ue(M), n E N, and a sequence of times tn ~ 00 such that Vn ~ v and cptn(vn) E Sl. As in section 2 this implies cP(cpt(v)) ~ O! for all t ~ 0 and there exists a sequence Sn ~ 00 such that Un = cpSn (v), n E N, is a Palais-Smale-sequence. After passing to a subsequence we may assume that Un ~ u als n ~ 00. Clearly cP(u) > (J because otherwise u E I, hence v E M contradicting Vn f/. Ue(M). The critical point u is an extremal sign changing solution. This follows as in section 2 using the almost connecting orbits cpt(vn), n E N.

References [1] Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18,620-709 (1976). [2] Ambrosetti, A., and Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14,349-381 (1973) . [3] Bartsch, T.: Critical point theory on partially ordered Hilbert spaces. Preprint. [4] Bartsch,T., Linking, positive invariance and localization of critical points. Preprint. [5] Bartsch, T., Chang, K.-C., and Wang, Z.-Q.: On the Morse indices of sign changing solutions of nonlinear elliptic problems. Math. Z.(to appear). [6] Bartsch, T. and Wang, Z.-Q.: On the existence of sign changing solutions for semilinear Dirichlet problems. Top. Meth. in Nonlin. Anal. 7, 115131(1996) . [7] Bates, P., and Jones, C.: Invariant manifolds for semilinear partial differential equations. in: Kirchgraber, V., and Walther, h.O.(eds.), Dynamics Reported, vol. 2, pp.I-38, Teubner(1989). [8] Castro, A.,Cossio, J., and Neuberger, J.: Sign changing solutions for a superlinear Dirichlet problem. Rocky Mountain J. Math. 27, 10411053(1997). [9] Castro, A.,Cossio, J., and Neuberger, J.: A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems. E.J,Diff.Eqn. 1998, 1-18(1998).

10

Thomas Bartsch

[10] Chang, K-C.: An existence of the Hess-Kato theorem to elliptic systems and applications to multiple solution problems. Preprint. [11] Chang, K-C .: Infinite Dimensional Morse Theory and Multiple Solution Problems, vol. 6 of Progr. Nonlin. Diff. Eq. Appl. Boston: Birkha.user [12]

[13] [14] ·[15]

[16]

[17] [18]

[19] [20] [21] [22] [23] [24]

[25]

1993. Cerami, G., Solimini, S., and Struwe, M.: Some existence results for superlinear elliptic boundary value problems involving critical exponents. J . Funct. Anal. 69, 289-306(1986). Conti, M., Merizzi, L., and Terracini, S.: On the existence of many solutions for a class of superlinear elliptic systems. Preprint. Conti, M., Merizzi, L., and Terracini, S. : Remarks on variational methods and lower-upper solutions. Preprint. Dancer, E.N., and Du, Y.: Competing species equations with diffusion, large interactions and jumping nonlinearties. J .Diff.Eqn. 114, 434475(1994) . Dancer, E.N., and Du, Y.: Existence of sign changing solutions for some semilinear problems with jumping nonlinearties at zero. Proc. Royal Soc. of Edinburgh 124A, 1165-1176(1994). Dancer, E.N., and Du, Y. : On sign-changing solutions of certain semilinear elliptic problems Appl. Anal. 56, 193-206(1995). Dancer, E.N., and Du, Y. : The generalized Conley index and multople solutions of semilinear elliptic problems. Abstract Appl. Analysis, l. 103-135(1996) . Dancer, E.N., and Du, Y.: A note on multiple solutions of some semilinear elliptic problems. J . Math. Anal. Appl. 211,626-640(1997) . H.Hofer, Variational and topological methods in partially ordered Hilbert spaces. Math. Ann. 261,493-514(1982). Krasnoselski, M.A.: Positive Solutions of Operator Equations. P. Noordhoff, Groningen, The Netherlands 1964. Krein, M.G., and Rutman, M.A.: Linear operators leaving invariant a cone in Banach space. Amer. Soc. Transl. 10, 199-325(1962). Li, S. and Wang, Z.-Q.: Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems. Preprint. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, AMS, CBMS Reg. Conf. Ser. in Math., 65, 1986. Wang, Z.-Q.: On a superlinear elliptic equation. Ann. Inst. H. Poincare, Anal. Non Lineaire 8, 43-57(1991).

ON THE PRINCIPAL EIGENVALUE OF ELLIPTIC SYSTEM WITH INDEFINITE WEIGHT AND APPLICATIONS Chang Kung-ching Institute of Mathematics, Peking University Beijing 100871, People's Republic of China

This is a survey paper on some progress in semilinear elliptic systems based on the simplicity or odd multiplicity of the first eigenvalue of the linearized system. cf. [C 2 ]-[C 5 ]. In the study of semilinear elliptic equations, the algebraic and geometric simplicity of the principal eigenvalue of a second order elliptic operator with indefinite weight plays an important role . For example, the bifurcation analysis, the construction of sub-and super-solutions, the linearized stability, and the computation of critical groups for a mountain pass point, see [HI], [HK], and [Cd. According to Krein-Rutman Theorem, the simplicity follows from the strong maximum principle, which implies the strong positivity of the resolvent of the second order elliptic operator. The ordered method based on the strong positivity, is very useful in the study of multiple solutions by combining with the topological degree theory or the critical point theory. For cooperative elliptic systems, a strong maximum principle was obtained by G. Sweers [S]. We first present an abstract version of Sweers' Theorem based on Krein Rutman Theorem. This theorem can be applied to periodic parabolic system as well. Then we turn to the first eigenvalue of the elliptic system with indefinite weight matrix. The Hess-Kato Theorem is extended. Various cases on the diagonal of the weight matrix are studied. In particular, a condition, which is related to the M-matrix, on constant weight matrices is introduced such that the extended version of Hess-Kato Theorem can be applied to cover many existence results in recent literatures, (see [FG], [FHT], [MMJ). Let n be a bounded domain with smooth boundary in ]Rn, let L = diag {Db D 2 , · · · ,Dp} where D k , k = 1,2,··· ,p, are second order uniformly elliptic 11

Chang Kung-ching

12 operators with continuous coefficients: Dk U

=-

n

L

a~7)(x) aij U +

i,j=l

n

L

b;k)(x) aj U + C(k) (X) U.

j=l

Let P,k be the first eigenvalue of Dk under the Dirichlet condition. We assume P,k 2: O. Given f € C 1 (IT x lRP , lRP ), we study the elliptic system:

L U = >. f (x, U) U=O

{ where U = (U1, ··· ,up). Let

M(x)

=au

in

n,

on

an,

(0.1)

f(x,O),

the linearization of (0.1) at 0 is the following eigenvalue problem:

LU=>.M(X)U U=O

{

in n on an

(0.2)

1. Strong Maximum Principle In the study of the Maximum Principle of elliptic systems: (L - B) U = V {

where B = (bij(x)) erative, i.e.,

€

·U=O

in n on an

(1.1)

C(IT,MI(p,lR)), it is naturally to assume that B is coop-

(1.2) > 0 if i 1- j cf [PW] . We use the notation: U 2: 0, if Uj(x) 2: 0 V j; U > 0 if U 2: 0 and if 3 k, 3 Xo € n such that Uk(XO) > 0; and U » 0, if 3 0: > 0 such that Uj(x) 2: o:ej(x) V j VuIT, where ej is the unique solution of Dju = 1 in n b·(x) 'J

with Dirichlet O-data, Vj. Maximum Principle says that if the homogenuous system (1.1) possesses a strict supersolution U > 0, then V > 0 implies U > 0, while the Strong Maximum Principle ensures U »0. It was G.Sweers [S], who introduced the fully coupled condition on B to ensure the Strong Maximum Principle. In his proof, a theorem due to de Pagter [P] on irreducible positive operators in a Banach lattice is applied in combining with the Krein-Rutman theorem. In the following, we shall present a more direct proof of this result. A matrix B € C (IT, M (p, lR)) is said fully coupled, if the index set {I, . . . ,

On the

y ............

"'.&l'~ .....&cs ....

.&.&t'c.w.u..:;;

v'&

0J0I.I":;;.&.&.I.

".&1.1.&.&

.&.&.&u.":;;'&'&'&'&..Ite

13

weight

p} cannot be split into two disjoint nonempty subsets I and J such that bij (x) == 0 V i €I, V j € J. In replacing de Pagter's Theorem, we need the following version of Krein Rutman Theorem (cf. [D)): Let X be a Banach space with a totally positive cone P. Let T be a positive compact operator in X satisfying the condition: VX

€

P:= P\{O}, 3 n

€

N such that (x*,Tn x) > 0

Vx· € P*\ {O}, where P* is the dual cone of Pin X'. Then (a) the spectrum radius reT) > 0 is a simple eigenvalue with eigenvector such that (x*, ¢) > 0 Vx· € P*.

q)f. P

(b) 3 ¢* € P* such that T* ¢*

= r (T) ¢* .

(c) IAI < reT) VA€a(T) with A =/:-r(T).

=

Now, let X = C (IT, IRP ) and let D j be the operator with domain D (D j ) {U€Co(IT)IDju€C(IT)}, then D j is closed, and ej(x) > 0 Vxd1, Vj . Let E (x) (el (x), ·· · , ep (x)), we have

=

(U* , E) > 0 V U*

€

p'.

(1.3)

Define

XE = U>.>o A [-E, EJ,

with norm

IlxilE = in! {A> 01 xd[-E,E]},

n. Ob-

where [-E, E] = {u = (Ul (x),· ·· , up (x)) € XllUj (x)1 :::; ej (x) Vx, V

viously, (X E , II . liE) is a Banach space continuously imbedded in X and possessing a positive cone PE = P n XE with int (PE) =/:- 0 in X E topology. (1.4) It is well known, that Vc > 0, the operator (cI +L)-l exists, and is a positive compact operator on X, where L = diag {D l ,··· , Dp} with domain D(L) = D(Dd x ... x D(Dp). Moreover, VVj€C(IT),Vj(x) ~ 0 but not identically equal to 0, we have a = a (Vj, c) > 0 such that (cl + Dj)-l Vj ~ aej V j. We write E j = ej (x) (0,··· ,1,···0), where 1 is in the j th component, Vj. According to the fully coupledness of B, we have (1.5) None of the direct sums Xi! E9 ... E9 X jk , 1 ::; jl < .. . < jk ::; p, are invariant subspaces of B, where Xj is the j th component of X. Since 3 Co > 0 such that Co I + B is strictly positive, We conclude that A = £";1 Be €.c (X, X) is strictly positive and compact Vc > Co, where Le = cl + L and Be = cl + B. We are going to show: 3q€[I,p] such that (1.6)

Chang l{ung-cnmg

14 Indeed, for V t: P,:1 j such that Uj > 0 then (i) (Be V)j = (c - co) Uj + (Beo V)j > 0,

(ii):1i -f:. j such that (BeV)i > O. For otherwise, BeVt:Xj,i.e., Xj is an invariant subspace of Be, and that is invariant to B . This contradicts with (1.5). Combining (i), (ii) and (1.4), we have AV ~ 01 (Ei + E j ) for some 01 > O. Repeating the above argument, :1 k -f:. i, j and 02 > 0 such that A2 V ~ 02 (Ei + E j + Ek) . After 1 ::; q ::; p steps, we arrive at Aq V ~ 0 E . Thus (V*, Aq V) ~ o(V*, E) > 0 provided by (1.3). From (1.6) and the above version of Krein Rutman Theorem, it follows T (A) > 0 and the existence of cJ> E P and cJ>. E p. such that A cJ> = T (A) cJ> and A* cJ>* = T (A) cJ> • . We claim T (A) < 1, if there exists V > (J such that (L - B) V>

Indeed, from Lc V > Befl, it follows Aq+1 V

(J

«

(J .

< A fl < fl, so is Aq V,

and then T

(A) (cJ>", Aq V)

Since (cJ>*, Aq fl) ~ If we write

0

=
*, Aq V

(cJ>*, E) > 0,

>= (cJ>", Aq+1 V) < (cJ>., Aq fl).

we proved :

T

(A) < 1.

L- B

= Le-

Be = L e (I - A),

B)-I

= (I -

A)-I L ; 1 =

then

00

T

= (L -

L

Aj L;I,

(1.7)

j=O

It is compact, and is strongly positive in the sense that 'V V > This is the Strong Maximum Principle.

(J, TV» (J .

Remark The strong maximum principle is extended to periodic- parabolic systems, cf [C 3 ]. For the scalar case, see [H2]. 2. Principal Eigenvalue From the strong positiveness of T, we have (V*, T V)

~ 0

(V", E) > 0

'V V" E

p.,

'V V E

p.

Again, by using Krein-Rutman Theorem, :1 p. > 0, \11 t: P and \11. (p. such that and

T· \11"

= p. \11",

15

On the principal eigenvalue of system with indefinite weight

or equivalently,

(L - B) l}I

= Jl- l

l}I

and

(L* - B*) l}I*

= Jl- l

l}I* .

Now, we return to problem (0.2) . The cooperativeness and the full coupledness of M (x) are assumed . If there exists a strict supersolution V>. > B of the homogenuous system(0 .2) , then there exist Jl (.\) > 0, l}I >. E P and l}I>' € P* such that

(L - .\ M) l}I >.

= Jl (.\)

l}I >.

and

(L * - .\ M*) l}I>'

= Jl (.\)

l}I>' .

Since Jl (.\) > 0, l}I >. is not a solution of (0.2). A natural way in approaching (0.2) is to introduce a parameter '"Y > 0, and to find a strict supersolution U>.>Bof(L-'\(M-'"YI))U=8. Letp(,\»O,l}I>. E P*,l}I>' E P·bethe associate eigenvalue problem solution of the operator L - .\ (M - '"Yl). Setting Jl (.\) = P (.\) - .\'"Y, this is

(L-'\M) l}I>.

= Jl(A)

l}I>.,l}I E P n D(L).

Then (0.2) is equivalent to solving a positive root of Jl (A) = 0. A strict supersolution V>. is constructed as follow: Find '"Y >

°

such that

p

L where 'l/Jj > V>. = ('l/JI, '"

mij 'l/Jj ~ '"Y 'l/Ji

V i,

j=l

°,'l/J

is the first eigenfunction of D j corresponding to p), we have

Jlj,

V j . Setting

(L-A(M-'"Yl)) U>. ?,diag{Jll,'" ,Jlp} U>. >B. In order to obtain a root of Jl, besides the cooperativeness and the full coupledness, we assume 3 Ao

>

03 U E D (L) n int (FE) such that (L - AoM) U E -int (FE)

(2.1)

It has been proved in [C 2 ] and [C 4 ] the following Theorem 2.1 Under the above assumptions, there exists a unique Al and ~ E D (L) n int (FE), ~* E D (£*) n int (FE) satisfying

= Al M

L ~

(ii)

dim ker (L - Al M) = dim coder (£* - Al M*)

(iii) odd,

~

and

L* ~*

= >'1 M*

(i)

>

°

~.,

= 1,

The algebraic multiplicity of All for the compact operator L -1 M is

Chang Kung-ching

16

(iv) V A > 0, if it is an eigenvalue of L U = AM U, then A 2:: AI. It is naturally asked whether All is algebraically simple. At this moment, it is not known. I leave it as an open problem. We present examples satisfying the condition (2.1). Ex.l

--

If max max mjj (x) > 0, then (2.1) holds. See [C 2 ).

un

I~j~p

For p = 1, this condition is also necessary for (2.1)'. However, for p > 1, since the cooperativeness plays a role, it is not necessary. For instance, p = 2,L = (-!:::,.) I, M =

AO > /-L1 (l-c)-l and U

= and det (BI) > 0, for all principle minors BI. By elementary linear algebra, if B is a M-matrix, then B- 1 exists and maps positive cone into itself. It was proved in [C 4 ) that if /-L1 > is the first eigenvalue of -!:::", then (2.1) holds. In this case AO = ,\-1 /-L1' More general examples can be constructed based on Example 2.

°

°

Ex.3 For an index subset 1 of {I, 2"" ,p}, we write !RI to be the projection space of !RP onto the components in I . Assume that L = (-!:::,.) 1, M (x) is continuous, cooperative and fully coupled, and that 3 Io C {I, 2,· ·· ,p} such that MIo is constant and fully coupled, and satisfies (Mo). Then (2.1) holds. Proof Let [(1 0 be the projection of the positive cone [( = !R~ onto !RIo' According to lemma 2.3 in [C 4 J, we find (10 E int ([(10) being a solution of the homogenuous equation (,\ - MIo)(Io = 0. And let (0 be the vector 'in !RP generated by (10 with zeroes in all other components, then (,\ 1 - M) (0

< ()

by virtue of the cooperativeness and fully coupledness of M.

On the principal eigenvalue of system with indefinite weight

Setting >'0 = we obtain

,.,,1/ J.. and Uo

17

> 0 is the first eigenfunction,

= (0 ® 'PI, where 'PI

(L - >'0 M) Uo < O. Choosing c > 0 such that c + mii(x) > 0 \:Ii \:Ixd"!, and letting Ac = AO (L + AO c I)-I (M + cI),

Again by the cooperativeness and the fully coupledness, we have

Uo < U1 < ... < Up

Up « U.

and

It implies

(L

+ AO cI) U = AO (M + cI) Up «

i.e., (2.1) holds. In particular, if

1101

AO (M

+ cI) U,

= 2, then the assumption reads as that :3 a submatrix

MIa which is constant, say

(~ ~),

in which ad:::; b c where a, d :::; 0 and

b, c > O. (However, if anyone of a and d is positive, then according to example 1, (2.1) is true).

3. Applications. 1. The construction of sub- and super-solutions We study the problem (0.1) with A = 1. U (and U) is called a sub- (and super- solution resp.) of (0.1) with A = 1, if

L U :::; f (., U)

(and

LU

~

f (., U)

resp.).

au

Since the elements on the diagonal of M (x) = f (x, 0) may not be all nonnegative, the ordered method can not be applied directly. We choose a positive number c such that c + mjj (x) > 0 \:I x E 0, \:I j, and let Ac be the first positive eigenvalue with eigenvector ~c Eint (FE) of the problem

Lc U

= A Mc

U.

(3.1)

We have lemma 3.1 Let Al be the first eigenvalue of the problem (0.2) . Then Al < 1 if and only if Ac < 1. In virtue of this lemma, the following theorem is a generalization of a result in equations .

Chang Kung-ching

18

Theorem 3.2 Suppose that the locally Lipschitzian function 1 € C (IT x IRP , IRP) is quasi-monotone, i .e., (j t-+ Ii (x; (1, .. . , (p) is monotone increasing for i f:. j, and that :3 M o, Moo € C (IT, M (p, IR)) satisfying the cooperative and fully coupled condition as well as (2.1). We assume that Ill(x , ~)-Mo(x)~II=o(II~11)

111 (x, O - Moo (x) Ell = 0 II~II

as as

II~II--+O , uniformly in x, II~II--+

00,

uniformly in x,

and that Al (Mo) < 1 < Al (Moo), where Al (M) is the first eigenvalue of (0.2). Then the system (0.1) with A = 1 possesses a strongly positive solution. The proof relies on the construction of a pair of strongly positive sub-and super-solutions. Firstly, from Al (Mo) < 1, we have Ac := Ac (Mo) < 1 and

. to ± int (P) as I>' - >'1 (M)I < ., u>.) E C±. Moreover, let ~o = ftr,~::J. 0, and let ~~ be the positive cone in ~p - Let g (x, ~O)

= lim

Edl~ [I (x, 0

-

M (x)~],

IEI--+oo

and

For 0 < I>' - >'1 (M)I < ., u>.) to C>., if (! (x, ~O) to int (PE ) \i ~o to ~~, I~ol = 1, then>. < >'1 (M); and if a (x, ~O) to - int (PE ) \i ~o to ~~, I~OI = 1, then>. > >'1 (M). The theorem extends a result due to Ambrosetti, Arcoya and Buffoni [AAB], where only the scalar equation with positive constant M is considered. 3. Mountain Pass Point Given a C 2 -function G on a Hilbert space H. Assume that G" is a compact perturbation of the identity. There are two definitions of a mountain pass point as follows: Let z be an isolated critical point of G. (Hofer [Ho]) For any small open neighbourhood N of z, Gc n N\ {z} is not path-connected, where c = G (z) and G c = {utOHI G (u) ::; c}. (Chang [C 1 ]) The first critical group C1 (G, z) ::J. o. It is proved in [BCW] that under the assumption:

0"

(z) ~ 0 and 0

to

u

(0"

(z)) imply dim ker Gil (z) = 1,

these two definitions are equivalent. Meanwhile, they are equivalent to rank Cq (G,z) = 0 is a const, 0 :s; 0: < 00 if n = 2, and 0 We consider the functional on HJ (0, JRP):

J (U) =

lemma 3.5

~

In IV' UI

2

dx -

:s;

0:


2.

In F (x, U (x)) dx.

(3.2)

Assume that U f HJ (0, JRP) is a mountain pass point of J, and

if

M (x)

= BE.

I (x, U (x))

is cooperative and fully coupled, then rank Cq (J, U) = Oql. Proof We have J" (U) = id - L- 1 M . It is easily shown that Al (M) = 1 if J" (U) ~ 0 and Of0' (J"(U)) . Noticing that LV = MV is equivalent to Lc V = Me V, for any c, in particular, we may choose c > 0 large enoul?,h such that c+mii(x) > 0 Vi, Vx . Theorem 2.1 is applied to show dimkerJ (U) = 1. As an application, it is shown in [C 2 ), (for scalar case see [DD]). Theorem 3.6

Assume (Ho) l(x,B) = B, Mo = BE. 1(', B), {O < J.LY < J.Lg :s; ... } be the positive eigenvalues of L w.r.t. Mo and J.Lg < 1. (Hoo) 3Moo fC(IT,M(p,JR)), which is cooperative and fully coupled and satisfies (2.1) . Let {O < J.L'r < J.L'f :s; ... } be the positive eigenvalues of L w.r.t. Moo, J.L'f < 1, and

III (x, €) - Moo (x) €II

= 0 (II€II)

as

II€II -+

00

uniformly in x,

(CF) B;jF(x,€»O Vi=J.j V(X,€)fOXJRP if p>1. In addition, if 0 E 0' (L - Moo), then there exist at least seven distinct nontrivial solutions of (0.1) with A = 1.

4. Ambrosetti Prodi type results. ([C s ]) It is a well known result in semilinear elliptic equations: if satisfies Q.

= lim sup I(t)

t-.>-oo

t

< Al < lim inf I(t) t-.>+oo

t

= ct,

I

E C(JR 1 )

On tbe principal eigenvalue of system witb indefinite weigbt

21

where Al is the first Dirichlet eigenvalue of -A on a bounded domain 0, and 'PI is the associated eigenfunction, then 3t* E IRI such that the equation

-Au = !(u)

+ t'Pl

u E HJ(O)

possesses no solution if t > t*, at least one solution if t = t*, and at least two solutions if t < t* . It is extended as follow: Theorem 3.7 Assume that! E CI-O(n, IRP) is quasi-monotone increasing and is of linear growth, and that A and A E C{n, M(p, IR)) satisfy: !(x,~) ~ A.(x)~

- Ceo,

(3.3)

!(x,~) ~ A(x)~ - Ceo,

(3.4)

where C > 0 is a constant and eo = (1,··· ,l)T. Assume (1) A. is cooperative and fully coupled, with (3.5)

(2) A is cooperative and fully coupled, with (3.6)

Given

O. The main theorem which we will prove is the following :

MAIN THEOREM

Let M3 be a three-dimensional complete non-

compact Riemannian manifold with bounded and nonnegative scalar curvature. Suppose M3 satisfies the Ricci pinching condition (1.4). Then the Ricci flow (1.1) has a solution 9ij(X, t) for all times 0

~

t < +00 and the curvature tensor

Rm(x , t) of 9i j(X , t) satisfies the following decay estimate (1.5)

where C is some positive constant. This paper is organized as follows. In Section 2, we give a Ricci pinching estimate for the Ricci flow. In Section 3, we prove the solution exists for all times 0

~

t < +00. In Section 4, we derive the decay estimate for the curvature.

2. The Pinching Estimate Let (M 3 , 9ij) be a three-dimensional complete, noncompact Riemannian manifold with bounded and nonnegative scalar curvature. Suppose (M 3 , 9i j)

28

Bing-Long Chen and Xi-Ping Zhu

satisfies the Ricci pinching condition (1.4). We evolve the metric gij by the Ricci flow

8gij(X,t) 8t {

gij(X, 0)

x E M 3 , t> 0,

(2.1)

=

gij(X),

From (1.4) we know that the Ricci curvature of (M3,gij) is nonnegative and then the Ricci curvature is bounded. Since the curvature

ten~or

is dominated

by the Ricci tensor in a three-manifold, we know from [19] the Ricci flow (2.1) has a solution on a maximal time interval [0, t max ) with t max

> o. The following

lemma shows that the Ricci pinching condition (1.4) is preserved as long as the solution exists and the Ricci pinching in improved when the scalar curvature is getting big.

LEMMA 2.1 t

= 0,

C
0 depending only on the initial metric such that (2.2)

Proof. Let us pick an abstract vector bundle V isomorphic to the tangent bundle T M. Choose an isometry u = {u~} between V and T M at the time t

= 0, and let the isometry u evolve by the equation (2.3)

The pull-back metric

The Ricci flow on complete three-manifolds

29

remains constant in time. Now we can use u to pull back the curvature tensor to V . Then it follows from [8] that (2.4)

oR at = ~R + 21 R cl

2

.

(2.5)

Hence

Since the metric gab is independent of time, the unit sphere Sp(l) of the fiber Vp at p is fixed. Define

P E M 3 , t E [0, t max ). This is,

then by a result of Hamilton (see Lemma 3.5 in [9]),

o

=

%t [ min (Rab - c:Rgab wESp(l)

f(p, t)gab)Waw b]

> Hence there exists wp E Sp(l) such that (2.7)

and (2.8)

Bing-Long Chen and Xi-Ping Zhu

30

On the other hand, by definition, we have (2.9) for any extension w of wp in a neighborhood of p with Iwl

= 1.

Particularly, if

we extend wp along geodesics emanating radially out of p by parallel translation to get a vector field in a neighborhood of p, the inequality (2.9) becomes

6. [(Rab - £R9ab - f(p , t)9ab)] (p) . w;wt ~

o.

(2 .10)

Combining (2.6), (2 .8) and (2 .10), we get (2.11) Diagonalise Rab at the point p such that Rab

= >"a9ab with

>"1

~

>"2

~

>"3.

Then wp is the eigenvector of >"1 and

1

R13l3

= 2"(>"1 + >"3 -

R 2323

= 2"(>"2 + >"3 -

1

>"2), >"1).

Thus by direct computations, the second term on the right hand side of (2.11) becomes

(>"1 h·

+ >"2 - >"3)>"2 + (>"1 + >"3 - >"2)>"3 - 2£(>"r + >..~ + >..~) f + A(>"~ + >..~ ) - 2B>"2>"3 ,

where h is some linear combination of >"1 , >"2 and >"3 , A = 1 - 2£ + 1~c

2£ ( _ 10 1-10

)2, B = _10 _ 1- 2£ ( _ 10 )2 1-10 1-10

-

The Ricci flow on complete three-manifolds

31

We may always assume that c ~ ~ which implies A ~ B. Then we obtain

8f(p, t) 8t ~ f:::.f(p, t) - C f(p, t),

for f(p, t)

~

0,

where C is a positive constant depending only on the bound of the curvature. Therefore we deduce that

f(·,t)

~ 0

by Shi's maximum principle for noncompact manifolds (e.g. see Theorem 4.6 in [20]). That says that (1.4) is preserved on 0 ~ t

< t max .

If (M 3, gij) is fiat, we have nothing to prove. So we may assume (M3, gij)

is not fiat. Thus it follows from the evolution equation (2.5) and the standard strong maximum principle that R

>0

for all t

> O.

Then without loss of

generality, we may always assume that R> 0 on M3 x [0, t max ) . To prove the second part, we consider

for small

a>

O. The function fu evolves by (see Lemma 10.5 and Lemma 10.7

of [8])

2(1-a) -8fu gpq\l p R\l qf u 8t = f:::./u + R -

2

--IR\l ·R ·k - \l '·R· R-kl R4-u 'J J

a~4__ :) (IRcI 2- ~R2) l\li R l2 + R32

_U

[a IRcl

2(I Rc I2- ~R2)

2

- 2P] , (2.12)

where P

~ c 21Rcl 2(IRcI2 - ~R2)

if Rij

~ cRgij

and R

> O.

Thus if we choose a ~ c 2 , (2.12) becomes

8fu 0 and a subsequence Pkj , tkj (j injectivity radius of the metric 9ij(·, t) at Pkj

for all j

Let us denote >.(P, t)

~

f..L(P, t)

~

= 1,2,·· · .

the

(3.3)

v(P, t) be the eigenvalues of the

curvature operator of the metric 9ij(·, t) at P . The scalar curvature R(P, t) =

>.(P, t)

+ f..L(P, t) + v(P, t)

is their sum. Since the curvature tensor at the time

= 0 is bounded, we may assume, without loss of generality, v ~ -1 at all points and at t = o. The long time pinching result of Hamilton (see Theorem t

4.1 in [15]) tells us that

R

~

(-v)[log( -v)

+ log(1 + 2t) -

3)

(3.4)

Bing-Long Chen and Xi-Ping Zhu

34

for any point P and any time t with v(P, t)

< O. This implies that when one

dilates the solution in the space-time so that the maximum curvature becomes 1, one can find a sequence 6j --+ 0 so that all eigenvalues of the curvature operator of the rescaled metric are at least -6j . Now by applying Theorem 25.1 in [13] to get the estimate (3.3), we only need to show that there exists a positive constant 6 such that (3.5)

for all k = 1,2, .. . .

Suppose not, then by (3.4) there is a subsequence, still denoted by Pk, tk(k

=

1,2, . . .), such that

(3.6)

as k --+ +00. Consider the marked evolving manifolds (M 3 , 9ij( " t), P k ), k

1,2,· · · For

each of these solutions, let Pk be the origin, translate in time so that tk becomes 0, dilate in space by a factor

0:

so that R(Pk, tk) becomes 1 at the origin at t

= 0,

and dilate in time by 0: 2 so that it is still a solution to the Ricci flow . The dilated solution exists at least on the time interval -tkRmax(tk) ::; t ::; O. Moreover by (3.1),(3.2) and Lemma 2.1, they satisfy a uniform curvature bound. In order to get a limit for the rescaled metrics, we adapt a trick of Hamilton (e.g. see the proof in Theorem 25.1 in [13]). For each k, we lift the domain in M3 which is the conjugate ball of the rescaled metric at Pk to its tangent space

R3 by the exponential map. It should be noted that such pull-back metric is only defined locally in space-time. Nevertheless, it is clear that the pullback metric still satisfies the Ricci flow equation. We also note the injectivity radius of the pull-back metric at the origin at the time t = 0 is greater than ~.

Thus we can extract a subsequence of the pull-back metric to get a limit

The Ricci Bow on complete three-manifolds

35

(defined locally in space-time) by the virture of the compactness theorem of Hamilton[12] . Moreover by (3.4) the limit has nonnegative curvature operator and by (3.6) the smallest eigenvalue of the curvature operator of the limit at the origin at the time t = 0 is zero. Using the strong maximum principle on the evolution equation of the curvature operator as in [9] (see Theorem 8.3 of [9]) we get that the smallest eigenvalue of the curvature operator vanishes on the whole limit, moreover, the null space of the curvature operator is parallel. Hence the holonomy reduces, which implies that the limit must be splitted locally as

~

x R for some nonflat surface

~.

Since the Ricci pinching condition

(1.4) is preserved under both evolution and dilation, we conclude that ~ x R must also satisfy (1.4). But it is impossible. Therefore the proof of the lemma is completed.

0

THEOREM 3.2

Suppose (M3, gij) is a three-dimensional complete

noncompact Riemannian manifold with nonnegative and bounded scalar curvature and satisfies the Ricci pinching condition (1.4). Then the Ricci Bow (1.1) with gij as initial metric has a solution for all times 0 ::; t < +00. Suppose the maximal existence time t max < +00, then

Rmax(t)

-+ +00 ,

as t -+ t max .

Let P k , tk be chosen in (3.1) and (3.2) . By Lemma 3.1, we can find a subsequence Pk;, tk; (j

= 1,2, .. .) and a constant 'Y > 0 such that (3.3) holds.

Applying the derivative estimates of Shi in [19], Lemma 2.1 and (3.1), we know that for each integer 1 ~ 0, there is a constant Cl depending on 1 such

Bing-Long Chen and Xi-Ping Zhu

36

that the lth covariant derivative of the curvature is bounded

far x E M3 j = 1,2, ···.

(3.7)

Dilate the metric at the time tkj so that

with the scalar curvature RU) (x)

=

1

R(Pkj , tkj)

. R(x, td

< 2,

, -

and

Then by (3.7) we get a sequence of marked Riemannian manifolds

(M3, g~P, Pkj ), j

= 1,2,· · ·,

such that for each nonnegative integer l there is

a constant Cl independent of j such that the lth covariant derivatives of the curvature

RW of the metric g~P satisfies a bound far all j

= 1,2, ... .

(3.8)

By (3.3) we know that there is a positive constant 'Y independent of j such that the injectivity radius of (M 3 , g~P , P kj ) satisfies

far all j

= 1,2,· ·· .

(3.9)

Hence by (3.8) and (3.9), the compactness theorem in [12] implies that there exists a subsequence of (M3, g~P, Pkj ) which converges to a complete marked manifold (M 3 , gij, Po) . Since every (M3,g~P,PkJ is noncompact, the limit (M 3,gij,PO ) is still noncompact. It is clear that

R < 2 on M3

and

R(Po) = 1.

(3.10)

The Ricci flow on complete three-manifolds

37

The pinching estimate (2.2) tells us that

RY) (X) 'J

~R(j)(X)g<j)(X)12 3 'J -::\;)

-

I Letting j

--t

< 22- Ck-:J fT

fT

.

9;j

+00, we have

~R(X)9ij(X)I;ii = °

IRij(X) -

This implies that the limit is a space form from the second Bianchi identity. Combining this fact with (3.10), we deduce that (M3,gij, Po) is compact. A contradiction holds. This proves the maximal existence time must be infinite.

o

4. The Decay Estimate In this section we will prove the following result. THEOREM 4.1

Let (M 3 , 9ij) be a three-dimensional complete non-

compact Riemannian manifold with nonnegative and bounded scalar curvature and satisfies (1.4). And let 9ij(X, t) be the solution gotten in Theorem 3.2.

Then the curvature tensor Rm(x, t) of 9ij(X, t) satisfies the following decay estimate for all x E M3 and t > 0,

(4.1)

where C is some positive constant. From Lemma 2.1 we know that the Ricci curvature of 9ij(X, t) is nonnegative for all t 2':

°

and all points in M3 . To prove (4.1) we only need

to show

IR(x, t)1 ~

C

t'

for all x E M3 and t> 0,

(4.2)

38

Bing-Long Chen and Xi-Ping Zhu

where C is some positive constant . Suppose not, then lim sup tRmax(t) =

(4.3)

+00.

t-too

We start by picking a sequence T j -+

+00

and a sequence 'Yj /' 1. Take

Pj E M3 and tj > 0 such that

Dilate the solution by

tj(Tj - tj)R(Pj , tj)

>

'Yj sup {t(Tj - t)R(x, t) ; x E M3 and t:::; T j

>

'Yj sup {t(Tj - t)R(x, t) ; x E M3 and t:::;

> '!if.-'Yj sup { tR(x, t) ; x E M3 and t
}

T} ¥'

00,

>

¥ (Tj~tj ) sup { tR(x, t) ; x E M3

-+

00,

and t:::; '!if.- }

> ¥(f;-)sUP{tR(x,t); xEM3 and t:::;~} -+

00 .

(4.4)

(4.5)

The Ricci flow on complete three-manifolds

39

By direct computations, we have (4.6)

for

tE

[-tj

/E; ,(Tj -

tj)

/E;) , and by (4.4), (4.5),

E;R(·,t)

< (4.7)

= -t

1,

as

j -t

00.

This says that we have a uniform bound on the curvature of the dilated metrics

~Jl (j = 1, 2, ... ). In order to extract a convergent subsequence, we need to get a uniform lower bound for the injectivity radius of ~Jl at Pj and at the time

t = O.

By noting the long time pinching (3.4) and (4.3), as well as (4.7),

we find that the same proof of Lemma 3.1 actually shows that the injectivity radius of the solution

gij (-,

t) at Pj and at t

= tj

satisfies

for all j

= 1,2, .. '.

where 'Y is some positive constant independent of j . Equivalently, the injectivity radius of the dilated metric ~Jl at Pj and at the time

t = 0 is

uniformly

greater or equal to 'Y for all j = 1,2,···. Thus we can use the compactness result of Hamilton[12] to extract a convergent subsequence so that the limit

(M 3 ,gij(·,t)) exists on the time interval

-00

"(Pjk ) - , Since the curvature operator of the limit metric is positive, we can use the Harnack inequality of Hamilton to conclude that R(x, t) is nondecreasing in time t. Then by the local derivative estimates in [13] and (i), we have

12 -2 (1) -1VRm(x,O) ::;CR(Pj,O) R(Pj,O)+r~

,

and then by (ii)

I~Rm(X,0)12

::; 2CR3 (Pj ,0),

for j large enough and x E Bri(Pj,O), (4.11)

where C is some positive constant. Set

Pj = VR(Pj,O)' where a is a small positive number to be determined. As k large enough, we get from (ii) that

and by (4.11) and (4.10), for x E Bpik (Pjk , 0),

iI(x)

> iI(Pjk ) - V2C. pjkR~(Pjk'O) > JA(Pjk ) - V2C. aR(Pjk,O) > (~- V2C. a) R(Pjk , 0)

>

(4.12)

42

Bing-Long Chen and Xi-Ping Zhu

by choosing

(7

= 12~ '

Thus up to a subsequence,

{Bp . (Pjk' O)}"" 'k

k=l

is a family of disjoint remote

curvature ,8-bumps for some ,8 > 0 in the sense of Hamilton[13]' which contradicts with the finite bumps theorem of Hamilton (Theorem 21.6 in [13]) .The contradiction proves the claim (4.9) . Consider the marked evolving manifolds (M 3 , gij(" t), Pj), j = 1,2,· ··. Dilate the metric gij (., t) as

Then we have another sequence of marked manifolds (M3, g}1\, t), Pj ), j

=

1,2, · .. , such that

(i)

iiU)(Pj,O) = 1, where iiU) is the scalar curvature of

P)· (M3 , ~U) gij' j ,

i3Y)(pj,O) is the ball of (M 3 ,g;J\.,0),pj ) with center at P j and

,

radius Tj; iiU) (x , 0)< 1 + 6J'·

(i-;)

the smallest eigenvalues vU)(Pj,O) of the curvature operator ii~) at Pj at the time t = 0 tends to zero as j -+ +00 (by (4.9));

(V)

each

YiP (.,t) is still a solution of the Ricci flow.

Again we use the local limit trick as in the proof of Lemma 3.1. For each j, we lift the domain in M3 which is the ball of (M 3 ,g;}\.,0),pj

)

with center at Pj

The Ricci flow on complete three-manifolds

43

and radius to be the conjugate radius of ~p (·,0) at Pj , to its tangent space R3 by the exponential map. Then we consider the pull-back metric on the lifted domain, still denoted by ~p

(.,0.

It is clear that the pull-back metric still

satisfies the Ricci flow equation.By (iii) we know that the injectivity radius of the pull-back metric ~p at the origin at the time

t=

0 is greater than ~ .

Thus by the virtue of the compactness theorem of Hamilton[12] we can get a limit (defined locally in space-time) from the pull-back metrics ~p

(-, 0.

It is

also clear that the limit of ~p still has nonnegative curvature operator and by

(i""V) the smallest eigenvalue of the curvature operator of the limit at origin at the time

t=

0 is zero. Again by using the strong maximum principle on the

evolution equation of the curvature operator of the limit as in [9] we get the smallest eigenvalue of the curvature operator vanishes on the whole limit and the null space of the curvature operator is parallel. This says the holonomy of the limit reduces. So the limit is splitted locally as I: x R for some nonflat surface I:. Since the Ricci pinching condition (1.4) survives into the limits, we deduce that I: x R must also satisfy (1.4). But it is impossible. Therefore the proof of the Theorem is completed.

o

Finally the main theorem is the combination of Theorem 3.2 and Theorem 4.1.

References 1. J .Cheeger and D.Gromoll, On the structure of complete manifolds of

nonnegative curvature, Ann. of Math. 96 (1972), 413-443.

44

Bing-Long Chen and Xi-Ping Zhu 2. B.L.Chen and X.P.Zhu, Complete Riemannian manifolds with pointwise

pinched curvature, Preprint. 3. B.Chow, The Ricci flow on the 2-sphere, J.Differential Geometry 71 (1991),325-334. 4. D.De Turck, Short time existence for the Ricci flow, J . Differential Geometry, 18 1983, 157-162. 5. G.Drees, Asymptotically flat manifold of nonnegative curvature, Differential Geom. Appl. 4 (1994),77-90. 6. J.Eschenburg, V.Schroeder and M.Strake, Curvature at infinity of open

nonnegatively curved manifold, J. Differential Geometry, 30 (1989),155166. 7. RE.Greene and H.Wu, Gap theorems for noncompact Riemannian man-

ifolds, Duke Math . J. 49 (1982), 731-756. 8. RS.Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry, 17 (1982),255-306. 9. RS.Hamiiton, Four-manifolds with positive curvature operator, J . Differential Geometry, 24 (1986),153-179. 10. RS.Hamiiton, The Ricci flow on surfaces, Contemporary Mathematics, 71 (1988),237-26l. 11. RS.:aamiiton, Eternal solutions to the Ricci flow, J.Differential Geometry, 38 (1993),1-11. 12. RS.Hamiiton, A compactness property for solution of the Ricci flow, Amer. J . Math. 117 (1995) ,545-572 . 13. RS .Hamiiton, The formation of singularities in the Ricci flow , Surveys in Differential Geometry, 2 (1995) pp.7-136,Intemational Press.

The Ricci flow on complete three-manifolds

45

14. R.S.Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom., 5 (1997), no.1, 1-92. 15. R.S .Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom., 16. G.Huisken, Ricci deformation of the metric on a Riemannian manifold, J.Differential Geometry, 21 (1985),47-62. 17. C.Margerin, Pointwise pinched manifolds are space forms, in proceedings of Symposia in Pure Mathematics, 44 (1986), 307-328 [Arcata: Geometric Measure Theory and the Calculus of Variations, July 1984). 18. S.Nishikawa, Deformation of Riemannian metrics and manifolds with

bounded curvature ratios, in proceedings of Symposia in Pure Mathematics, 44 (1986),343-352 [Arcata: Geometric Measure Theory and the Calculus of Variations, July 1984). 19. W.X.Shi, Deforming the metric on complete Riemannian manifold, J . Differential Geometry, 30 (1989),223-30l. 20. W.X.Shi, Ricci deformation of the metric on complete noncompact Rie-

mannian manifolds, J.Differential Geometry, 30 (1989),303-394.

Heteroclinic Orbits of Second Order Hamiltonian Systems Chao-Nien Chen Department of Mathematics National Changhua University of Education Changhua, Taiwan, ROC Abstract We study a class of second order Hamiltonian systems. Variational arguments and penalization methods are used to obtain the existence of heteroclinic orbits joining pairs of equilibria.

Consider the Hamiltonian system (HS)

ij - V'(t,q)

= 0,

where q : JR -t JRn, V E C 2 (JR x JRn,JR) and V'(t,y)

= DyV(t,y) . The

basic

assumptions for the function V(t,y) are the following:

(Vl) There is a set Kl C JRn such that if f/ E Kl then V(t, f/)

= yERn inf V(t , y) =

Vo for all t E JR. (V2) There are positive numbers J.Ll, J.L2 and Po such that if Iy - f/I ::; Po for some f/ E Kl then J.L21y - f/12 ::::: V(t,y) - Vo ::::: J.Llly - f/12 for all t E IR.

46

Heteroclinic orbits of second order Hamiltonian systems

47

+ 11-0

for some t E JR then

(V3) There is a 11-0 > 0 such that if V(t, y) ::; Vo

Iy -

1]1 ::;

(V4) For any

Po for some

TO

1] E

K1·

> 0 there is an M > 0 such that sup II D;V(t,y)

1100::;

M if

tEIR

IYI :::; TO · By (VI) , any element of Kl is an equilibrium of (HS).

W

seek a solution q

of (HS) which satisfies lim q(t) =

t-+- oo

for a pair of 1]i , 1]j

E K 1.

1]i

and lim q(t) = t-+oo

(1)

1]j

Such a solution will be called a heteroclinic orbit of

(HS). If V is periodic in t and in each component of y, Strobel [St] showed that, for

any

1]i E

K 1 , there is a heteroclinic orbit q of (HS) which satisfies q(t) -+

t -+

-00

and q(t) -+

Kl \ {1]d

as t -+

00.

Moreover, for any pair of 1]i, 1]j

1]i

as

E K 1,

they can be joined by a chain of heteroclinics. If additional nondegenercy conditions are satisfied, there exist multi bump heteroclinic orbits connecting 1]i

and

1]j.

Such kinds of results have also been proved by Rabinowitz [RI] and

Maxwell [MI ,M2] for orbits connecting periodic solutions instead of equilibria. Our aim in this paper is to investigate the heteroclinic orbits of (HS) when

V is not a periodic function. To avoid complicated notation, we restrict our attention mainly to the case of

/(1

= {1]1, 1]2}. Let

function which satisfies Q(t)

= { 1]1

1]2

if if

Q E C 2 (JR, JRn) be a fixed

Chao-Nien Chen

48

By (HS), the potential V is only determined up to an additive constant, so we may assume that Vo

For

Z

E

If IQ(z)

= O. Let E = W 1•2 (]R, ]Rn) with the norm

E, define

= 0 and IQ(z) > 0 then the function q(t) = Q(t) +z(t) is a heteroclinic

orbit of (HS) . Let

Q:

= infzEEIQ{z). A sequence {zm} C E is called a (PS)c sequence of

IQ if IQ(zm) -+ c and IQ(zm) -+ 0 as m -+ a minimizing sequence if IQ(zm) -+

Q:

In particular {zm} is called

00 .

as m -+

00.

A difficulty arised in the

study of variational problem on unbounded domain is that the Palais-Smale condition may not be satisfied (see e.g., [CES, CR, dFI, DN, L, MI, M2, S)) . Our approach is to search critical points of IQ by investigating the convergence of Palais-Smale sequences. The investigation will be based on a comparison argument described as follows. For kE N, let Ek = {z E Elz(t) if t ::; k} and E-k

Theorem 1.

= {z

E Elz(t)

+ Q(t) = 1]2

+ Q(t)

if t ~ -k}. Define

= Q:k

1]1

=

If there is a kEN such that (2)

then there is a function q which satisfies (HS) and lim q(t)

t-t-oo

= 1]1 .

lim q(t)

t-too

= 1]2.

(3)

Heteroclinic orbits of second order Hamiltonian systems

Remark. A simple example for which (2) holds is t~~ (t,y)

49

> 0 for t 1= 0 and

y ¢ {1Jl, 1J2}. Before proving Theorem 1, we state several technical results.

Proposition 1.

Let P E (0, Pol and fJ(p)

8Bp(1Ji), q(t2) E 8Bp(1Jj) and q(t) E lR n\(

= min(/LIP2, /Lo) . Suppose q(td E U B p(1J)) for t E (tl,t2). If i 1= j

1)EJC 1

then

Lemma 1.

Let {zm} C E be a (PS)c sequence. Then there is a constant

Co> 0 such that sUPmEN II zm 11L2(IR)~ Co· Lemma 2.

If {zm} is a (PS)c sequence then {Zm} is bounded in LOO(lR, lRn).

Proof of Theorem 1.

Let {zm} be a minimizing sequence. By Lemma 1 and

Lemma 2, {zm} is bounded in Wl~';(lR, lRn) . Hence there is a Z E WI~';(lR, lRn) such that along a subsequence L~c(lR,

Zm

-+

Z

weakly in WI~'; (lR, lRn) and strongly in

lRn). It follows that IQ(z) ~ o.

Let q = z+Q. We are going to show that q is a heteroclinic orbit of (HS) and

= {t E lRld(t) < p} and Bp = lR\Sp. It follows from Proposition 1 that IBpl < 2l(p), where l(p) = [29(pjl + 1 and IBpl is satisfies (3). Let d(t)

= 1)EJC inf

Iq(t)-1JI, Sp

1

the Lebesgue measure of Bp. Since

0

< min(O_k' Ok),

direct calculation shows

Chao-Nien Chen

50

that there are Pl , P2 E (O,Po) such that q(t) E and q(t) E Sp

Bp2

(17d if t E

SP2

(172) if t E

BpI

n (-00, -k - 1) . Set p

=

SPI

n (k + 1,00)

min (PI ,P2), Al

=

n (-00, -k - I), A2 = Sp n (k + 1,00) and A3 = lR\(Al U A2) ' It follows

from (V2) that

By Proposition I,

as

IA31

~ 2(l(p) +k+ 1) . This implies that z E E. Thus z(t) -+ 0

It I -+ 00 and q satisfies (3) . Next, we study the existence of multiple heteroclinic solutions of (HS) . Let 1

2

B =

UB

po

(17i) and A =

i=1

1IV'(t, y)11 +2'

sup (t ,y)ElR x B

E(jl,h) = {z EEl z(t)+Q(t) = 171 if t ~ jl

and

z(t)+Q(t) = 172 if t 2: h}.

Similarly, &(k,oo) = infzEE;(k,oo) IQ( z ) and &(-00, k) = infzEE;(_oo,k) IQ(z ), where

E(k , 00) = {z E Elz(t)

+ Q(t) =

171

if t ~ k}

and

E(-oo,k)

Theorem 2.

= {z E Elz(t) + Q(t) = 172

Suppose there are kl

if t 2: k} .

< k2 < k3 < k4 such that &(kl' k2)
0 such that y . V'(t,y + Q)

~ elYl2

if (t,y) E ((-oo,-kj x BR1(0)) U ([k,oo) x BR2(0)). (V6) There are

(h,

()2,

R 3 , R 4 , k E (0,00) such that

Theorem 4. Assume (V5), (V6) and the hypotheses of Theorem 3 are satisfied.

If 13

< min(a(-00,k 1 ),a(k4 ,00)) and max(j3 - IQ(q - Q),j3 - IQ(ij - Q))
2. Let

Qi,j E C2(1~,

JR.n) be a fixed function which satisfies

Qij(t) = { "Ii , T}j

if if

Define

cr(i,j)

= zEE inf IQ, (z) . t,

54

Chao-Nien Chen

For kEN. let

and

Define

It is easy to check that

Ok(j, I)

=

}nf

zEEk(i,j,l)

[~IQi,j + 2:12 + V(t, Qi ,j + Z)] dt. iroo k

Similarly, we define

o_k(i,l) = where

Set

and

inf

qEE_.(i ,l)

j

-k

-00

1

[-iq12 + V(t,q)]dt, 2

Heteroclinic orbits of second order Hamiltonian systems

55

Theorem 5. If there is a kEN such that a(i,j) < min(Q_di),Qk(j)), then there is a function q which satisfies (HS) and (1) . The proof of the theorem is similar to that of Theorem 1. Acknowledgments: The author would like to thank Professor Yiming Long for his invitation to the conference and the members of the Nankai Institute of Mathematics for their gracious hospitality.

References [AB]

A. Ambrosetti and M. Badiale, Homoclinics: Poincare-Melnikov type results via a variational approach, Comptes rendus de L' Academie des Sciences, Ser. i, mathematique, 323 (1996), 753-758.

[AR]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349381.

[BCT]

B. Bufi'oni, A. R. Champneys, and J . F . Toland, Bifurcation and coalescence of a plethora of homo clinic orbits, J . Dyn. Difi'. Eq., 8 (1996), 221-279.

[BS]

B. Bufi'oni and E. Sere, A global condition for quasi-random behavior in a class of conservative systems, Comm. Pure Appl. Math., 49 (1996), 285-305.

[CES]

V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homo clinic orbits in Hamiltonian systems, Math. Ann. 288 (1990) , 133-160.

[CR]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J .A.M.S. 4 (1991), 693-727.

[CT]

C. -N. Chen and S. -Yo Tzeng, Existence and multiplicity results for heteroclinic orbits of second order Hamiltonian systems, J. Differential Equations, in press.

56

Chao-Nien Chen

[dFl]

M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. 4 (1996), 121137.

[DN]

W .-Y. Ding and W .-M. Ni, On the existence of a positive entire solution of a semilinear elliptic equation, Arch. Rat. Mech. Anal. 91 (1986), 283-308.

[F]

P. Felmer, Heteroclinic orbits for spatially periodic Hamiltonian systems, Ann. Inst . H. Poincare - Anal. Nonl. 8 (1991), 477-97.

[KKV]

W . D. Kalies , J . Kwapisz and R. C. A. M. VanderVorst, Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria, Comm. Math. Phys. 193 (1998), .337-371.

[L]

P. L. Lions, The concentration-compactness principle in the calculus of variations, Rev . Mat. Iberoam. 1 (1985), 145-201.

[Ml]

T. O. Maxwell, Heteroclinic chains for a reversible Hamiltonian system, Nonlinear Anal. 28 (1997), 871-887.

[M2]

T . O. Maxwell, Multibump heteroclinics for a reversible Hamiltonian system, Preprint .

[Mo]

P. Montecchiari, Multiplicity of homo clinic solutions for a class of asymptotically periodic second order Hamiltonian systems, Rend. Mat. Ace. Lincei s.9, 4 (1994), 265-271.

[Rl]

P. H. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergod. Th. and Dynam. Sys., 14 (1994), 817-829.

[R2]

P. H. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, 2, Differential and Integral Equations, 7 (1994), 1557-1572.

[R3]

P. H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Var. 1 (1993), 1-36.

[R4]

P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system. Ann. Inst. H. Poincare, Anal. Nonlineaire, 6 (1989) , 331-346.

Heteroc1inic orbits of second order Hamiltonian systems

57

[R5]

P. H. Rabinowitz, Multibump solutions of differential equations: An overview, Chinese Journal of Mathematics 24 (1996), 1-36.

[RT]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z. 206 (1991), 473-479.

[S]

E. Sere, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z. 209 (1992), 27-42.

[St]

K. Strobel, Multibump solutions for a class of periodic Hamiltonian systems, University of Wisconsin, Thesis, 1994.

On the connecting orbits among local minimizers for monotone twist map* Wei Cheng and Chong-Qing Cheng Department of Mathematics, Nanjing University 210093, Nanjing, P. R. China

Abstract In a Birkhoff region of instability for an exact area-preserving twist map, we construct some orbits connecting distinct Denjoy minimal sets and the non-birkhoff periodic orbits, these sets correspond to the local, instead of global minimum of the Lagrangian action. In the earlier work of Mather [M2J, he constructed connecting orbits among Aubry-Mather sets, the global minimizer of the Lagrangian action.

1

Introduction

Consider an exact area-preserving twist diffeomorphism I of the cylinder IR/Z x R We let O(mod 1) denote the standard coordinate of IR/Z and let y denote the standard coordinate of IR. By a Birkhoff region of instability we mean a compact I-invariant subset of the infinite cylinder, which has two invariant closed curves r _, r + as its boundaries, each homeomorphic to a circle noncontractible to a point, and there is no other noncontractible circles. Let p(r) denote the rotation numbers of f at r, due to the twist condition we have p(r _) < p(r +) ( or vice versa), and each circle must be the graph of some Lipschitz function. Although I has some topologically dynamical stability in a fixed Birkhoff region of instability, it exhibites very complicated dynamical behavior indeed, much of them remains unknown still. Nevertheless, some remarkably • Supported by NNSF of China

58

On the connecting orbits

59

profound progresses have been made in the past two decades, among them are Aubry [A] and Mather's work [MI, M2, M5]. In [A] and [MI], Aubry and Mather proved respectively that for any rotation number w there exists a I-invariant Cantor set, I is wquasi-periodic on this set. Such set corresponds to the global minimum of Percival's Lagrangian. In 1985, Mather extended the result discovering that for any irrational number w, there exist uncountable many I-invariant sets. Such sets correspond to local, instead of global minimum of the Lagrangian. Another developement occur in 1991, Mather gave a result on the connecting and chaotic orbits among Aubry-Mather sets in [M5]. He obtained that between any two distinct Aubry-Mather sets there exists a connecting orbit. Moreover, Mather constructed some "stochastic" orbits of a series of Aubry-Mather sets M Wi , i E Z, i.e., such orbits can move arbitrarily close to each M Wi • In this paper we shall firstly extend Mather's work on Denjoy minimal sets [M2] to the rational case, i.e. , the existence of non-Birkhoff periodic orbits, then we shall extend Mather's work on connecting and "stochastic" orbits, to prove that such kind of orbits also exist among Denjoy minimal sets and non-Birkhoff periodic orbits, corresponding to local minimum of the Lagrangian. The proof is based on variational argument, and has two key steps: First, by a relative minimizer 0, and 11.6.11 < 8(h,w,n,Pw(a)), then the inequalities 2) and 3) in the definition of Ana~ are strict for ¢ = ¢wna~, where Fw is the modified Percival's Lagrangian and Pw(a) is the Peierls' barrier with respect to w and a. LEMMA

When w is irrational, a minimal elements of the modified Percival's Lagrangian Fw over Ana~ determines a Denjoy minimal set Mna~w, the closure of the set (¢wna~(t), 1Jwna~(t)). Mather proved that for fixed w and a, Mna~w and Mna~/w decides two distinct Denjoy minimal sets unless ,6. equals to .6.' modulo a translation, then there exist uncountable many I-invariant Denjoy minimal sets for the same irrational rotation number wand a E ~ where Pw(a) > O.

On the connecting orbits

3

63

Existence of non-Birkhoff periodic orbits

In this section we shall show that is for some irrational number w the Peierls' barrier Pw(a) > 0 then there exists non-Birkhoff periodic orbits for rational number p/q -+ w with q large enough, which is the assertion of the following theorem. THEOREM 1 Let f be an area-preserving monotone twist map of the infinite cylinder imposed on Mather's additional regularity conditions (Hs) and (H66) as well as those mentioned previously. Let w E JR \ Q be a rotation number such that there is no invariant circle of rotation number w, i.e., Pw(a) > 0 for some a E JR, then there exist infinite many non-BirkhoJJ periodic orbits with the same rational rotation number sufficiently close to w.

Similar to Mather's proof of the existence of Denjoy minimal sets, we also need to prove If Ana~ is nonempty, then there exists ¢>wna~ E Bna~ which minimizes Fw over Ana~ when w is a rational number. If w is irrational, Pw{a) > 0, 11.6.11 = maxi,jEI.6.{i) - .6.(j)1 is sufficiently small, then there exists a rational number p/q sufficiently close to w such that the inequalities 2) and 3) are strict for ¢>l!.na~ ' More precisely, we have that q if ¢> E Bna~ minimizes Fw over Bna~, then LEMMA 3

2')

¢>(t) < a + j, ift ~ j

¢>(t) > a + j, for any j E Z. 3')

+ .6.(j), if t > j + .6.(j)

The proof of lemma 3, consists of several steps. The first one is the procedure of some approximations based on the following two facts. One is a basic fact in Diophantine approximation theory and the other is an estimate on the continuity of modulus of Peierls' barrier by Mather. It is well known that for any irrational number w, there exist arbitrary accurate approximations whose error is less than the reciprocal value of the square of the denominator, i.e., Iw - p / ql < 1/ q2 . In fact

64

Wei Cheng and Chong-Qing Cheng

such a rational approximation can be precisely chosen as the continued fraction expansion of w.

4 ([M4]) If h is the continuous real valued function satisfying (Ht}-{H68), and Pw{a) denote Peierls' barrier with respect to h, then

LEMMA

IPw{a) - P!'.{a)1 q

~ (1200)~q (1 + Iw -

EI) q

for any a E R p/q E Q expressed in lowest terms and w E S'the space of rotation symbols, where () is a constant dependent only of h. So we can choose rational numbers p/q close to irrational w such that the Peierls' barrier Pd a) is sufficiently close to Pw(a), then the q condition Pw{a) > 0 leads to Pda) > O. q

If 0, then

LEMMA 5

and

II~II

o ~ nio - 1 + ~+ for II~II similarly. we choose t = s - ~+, then

+ wio t + wio

t

+ wio < nio' ~+ + wio > nio -

= s - ~+ = s-

Together with Case 1, we have that ift K j .• Since K

K(w, a), Proposition 2 and Lemma 1 above implies

10 Let .1 = ( .. . ,Ji, .. . ) be a constraint and w, a be real numbers. Suppose that J i = [ai, ai + 1] for] - K ~ i ~ ] + K with ai defined as above. Let ~ = ( ... '~i' . .. ) be a .1 -minimal configuration, then ~ is free at] if II~II < min{ 8(h, w, n, Pw(a)), Ilqm-lWII}· LEMMA

With these preliminary work we can construct the connecting orbits of Denjoy minimal sets now. The key step is to define a constraint .1 = ( ... , Ji, . .. ). Consider the constraint .1* = ( ... , Jt, ... ) defined in [M5] to construct the connecting orbit of Mather sets MWI and MW2 in the fixed Birkhoff region of instability. There exists integers ]1 and 12 such that if i E Kjl there exist tl E ~ such that ¢Wl (tl + WI i) E Jt, and if i E Kh there exists t2 E ~ such that ¢w2(t2 +Wli) E Jt- Here tl and t2 can

On the connecting orbits

75

be perturbed slightly such that tl + ~± + wli - ~ (k) f/. Z for i, k E Z and 1 = 1,2. From Lemma 1, we see that there exist two relatively minimal configurations ( . .. , CPt::..l (SI + Wli), ... ) and ( . .. , CPt::..2(S2 + W2i), . .. ) such that for i in Kjl and Kh they determine the same constraint as the corresponding minimal configurations do. Then we define the constrain .Jas [ai,ai + 1] for i:::;; ]li Ji = Jt for ]1 < i < hi { [a~, a~ + 1] for i ~ 12, where ai-a and a~-a' are all integers such that CPt::..l (SI +Wli) E (ai, ai+1) for i E Kh and CPt::..2(S2 +W2i) E (a~,a~ + 1) for i E K h . Thus we have

If the constraint .J is defined as above with PW1 (a) > 0 and PW2 (a') > O. Suppse that p(f _) :::;; WI, W2 :::;; p(f +) with WI irrational for 1 = 1, 2, 1I~111 < min{ 8(h, WI, nl, a), Ilqml-lWdl} and 11~211 < min{8(h,w2,n2,a'),llqm2-1W211}. Let ~ be a .J-minimal configuration. Then ~ is .J -free and corresponds to an I-orbit. PROPOSITION 2

PROOF. We only need to verify the .J-freeness at each i E Z. For ]1 :::;; i :::;; 12, that is from Mather's arguements in [M5]i Lemma 2 is appied otherwise. _

Now the statement above guarantees the existence of such an I-orbit, then we want to prove that it is exactly the connecting orbit we want.

5

Close approach

Note that if x = ( ... , Xi, . . . ) and y = ( ... , Yi, . .. ) are defined as Xi = CPt::..(t + wi) and Yi = CPt::..(s + wi) with t < s, then X < y, i.e., Xi < Yi for all i E Z since CPt::..(t) is strict increacing function. So the relatively minimal configurations defined by CPt::.. with the same rotation number and the same restriction ~ has total order. Recall that the Aubry graph of a configuration z = ( ... , Zi, ... ) is the broken curve in the plane that is the union of the line segments joining (i, Zi) and (i + 1, Zi+1). Now we will prove a relative Aubry crossing lemma.

Wei Cheng and Chong-Qing Cheng

76

11 (RELATIVE AUBRY CROSSING LEMMA) Let.J = ( ... , h ... ) be a constraint such that Xi E Ji, where Xi = ¢tl(s + wi) with s + wi~(k) ¢ Z for i, k E Z. Let Z = (... ,Zi, .. . ) is also defined by ¢tl, i.e., Zi = ¢tl(t + wi). If there exists two indices io, il such that Zio > aio + 1 and Zil > ail + 1 (resp. Zio < aio and Zil < ail ), then the Aubry graph of Z and of the .J -minimal configuration ~ never cross in the range

LEMMA

io~i~il' PROOF.

We have h((~ 1\ Z)io"'" (~ 1\ Z)il) ~

+ h((~ V Z)io"'"

(~V Z)il)

+ h( Zio' ... , Zil )

h( ~io' ... , ~il)

by (H5) ' Here (~ 1\ Z)i = min{~i' zd, (~ V Z)i = max{~i' zd. Equality holds if and only if all crossings of the Aubry graph occur at nodes. If we do not have equality, then we have the following comtradictions. Since ~ 1\ Z is a .J-configuration that agrees with ~ at i = io, iI, and for the configuration ~ V Z there exist 'lj; E Anatl such that (~V Z)i = 'lj;(t + wi), i.e., (~ V Z)i E Xwntlt± for io ~ i ~ i l and agrees with Z at i = io, iI, then either

or h((~ V Z)io," " (~V Z)il)

< h(Zio'''' ,Zil)

holds, which contradicts to the minimal assumptions on Z and~. The construction of such 'lj; is as follows: let a > 0 be small and set 'lj;(s) = (~V Z)i 'lj;(s

+ n)

'lj;(s)

if io ~ i ~ iI, t

= 'lj;(s)

= ¢tl(s)

+n

+ wi

~ s ~ t

+ wi + a;

for all s;

otherwise.

Since w is irrational so for io ~ i ~ il such 'lj; exists. If we have equality, then

h((~ I\Z)io''' '' (~ I\Z)il) h((~VZ)io,,,,,(~VZ)il)

=

h(~io''' ' '~il)' h(Zio, .. ·,Zil)'

On the connecting orbits

77

But if the graph cross at i, it contradicts to (H4)' • Now we have the following lemma on the close aproach. LEMMA 12 Let w be an irrational real number and a E ~ be such that Pw(a) > O. Let Y < x < Z be relatively minimal configurations of rotation number wand the restriction~. For each i E Z, let ai be the unique real number such that ai - a E Z and Xi E (ai, ai + 1). There exists an integer K > 0 such that if J = (... ,Ji,"') is a constraint with Ji = [ai, ai + 1] for j - K :::;; i :::;; j +K, then any J -minimal configuration ~ = ( .. . , ~i' ... ) satisfies Yj < ~j < Zj. PROOF . If K is large enough chosen as before, there exists i o, i l that satisfy j - K :::;; io :::;; j :::;; il :::;; j + K such that ai + 1 < Zi for i = io, il' ( In fact if x and Z are defined by Xi = ¢6.(s + wi) and Zi = ¢6.(t + wi) with real numbers 8 < t, we define intervals Wi = (8, t) + wi which can cover some io + ~(io) if K > K(w, a). ) Since ~ is a J-configuration, we have ~i < Zi for i = io,i I . Then from the relative Aubry crossing lemma above, ~i < Zi for io :::;; i :::;; i l . The proof of Yi < ~i is similar .• PROOF OF THEOREM 1. As we mentioned previously, if Y = (... , Yi, .. .) and z( ... ,Zi,"') are two relatively minimal configuration defined by the same ¢6., they have trichotomy: Y < z, Y = Z or Z < y, where Y < Z means that Yi < Zi for each i E Z. Since then, if Y < z then the projection of the intevals fYi, Zi], i E Z on ~/Z do not overlap, which means that Y and z are a-asymptotic and w-asymptotic. With Lemma 4, the statement of our main result is proven .•

6

"Stochastic" orbits

As mentioned previously, the construction of "stochastic" orbits among Denjoy minimal sets is simila,r to that of connecting orbits. Given an irrational number Wk for each k E Z we construct the "stochastic" orbits as follows: consider a series of Denjoy minimal sets {Mnkak6.kwkheZ, and a series of constraints {Jk}keZ' each Jk is used to construct the connecting orbits between the Denjoy minimal sets Mnkak6.kwk and Mnk+lak+16.k+1Wk+1' if for each k E Z the Peierls' barrier PwK(ak) > 0

78

Wei Cheng and Chong-Qing Cheng

and II~kll < min{6(h,wk,nk,ak), Ilqmk-IWkll} where each nk E Nand ~k E 'Dnk . Now for any k, there exists an integer jk such that the .Jkminimal configuration ~k satisfing l4>t (Sk + wdk) - ~jk I < 10k by Lemma 4 with small 10k > 0 for each k. For the coincidence of the segments of constraints, we need to show Jjk = Jlk- l . In fact, if jk is increasing with respect to k and jk - jk-I is very large, we can choose jk such that two constraints .Jk and .Jk- I coincide at jk by the definition of the constriants for constructing the connectiong orbits. Then we define a constraint .J = ( ... , Ji , .. . ) by

for k E Z. Then we have the following proposition on the existence of the "stochastic" orbits among a series of Denjoy minimal sets. THEOREM 3 Let 4>wknkak!:>.k is the minimal elements of the modified Percival's Lagrangians FWk over Ankak!:>.k for each k E Z. Suppose that Wk is irrational with p(f _) ~ Wk ~ p(f +) and the Peierls' barrier PWk (ak) > 0 with ak E IR and k E Z. For ~k E 'Dnk , k E Z with qmk defined previously, we also suppose that II~kll < min{ 6(h, Wk, nk, PWk (ak)), Ilqmk-IWkll} for each k. Given a series of real numbers {Ek}kEZ, then there exists an /-oribts 0 such that 0 can 10k-close to Denjoy minimal sets Mnkak!:>.kwk for each k. PROOF. It is a easy consequence of the arguments for the connecting orbits, Lemma 4 and the construction of .J.• REMARK 3 It easy from the construction above to construct the connecting orbits and "stochastic" orbits of the non-Birkhoff periodic orbits in the similar way if the denominator q of the rotation number p / q is large enough. Wang Qiudong [WI, W2] shows that for local minimal fixed points some connecting orbits exist. But whether there in general exist connecting orbits between the local minimum in the rational case when the denominator of the period p/q is not very large is still open.

On the connecting orbits

79

References [A]

Aubry, S., The twist map, the extended Frenkel-Kontorova model, devil's staircase. Physica D, 8(1983), 240-258.

[A-L] Aubry, S. and Le Daeron, P. Y., The discrete Frenkel-Kontorova model and its extensions. Physica D, 8(1983), 381-422.

[Ba]

Bangert, V., Mather sets for twist maps and geodesics on tori. Dynamics Repoted I, 1988, 1-45.

[Bi]

Birkhoff, G. D., On the periodic motions of dynamical systems. Acta Math., 50(1928), 359-379.

[BH] Boyland, P L. and Hall, G. R., Invariant circles and the order structure of periodic orbits in monotone twist maps. Topology, 26(1987), no. 1, 21-35. [CC] Cheng, W. and Cheng, C.-Q., Existence of infinite non-BirkhofJ periodic orbits for area-preserving monotone twist maps, Sci. in China, Chinese version, 29(1999), no. 12.

[Ml] Mather, J . N., Existence of quasi-periodic orbits for twist homoemorphism of the annulus. Topology, 21(1982), 457-467.

[M2] Mather, J. N., More Denjoy invariant sets for area preserving diffeomorphisms. Comment. Math. Helv., 60(1985), 508-557.

[M3] Mather, J . N., A criterion for the non-existence of invariant circles. Publ. Math. I. H. E. S., 63(1986),

[M4] Mather, J. N., Modulus of continuity for Peierls's barrier. Periodic solutions of Hamiltonian systems and related topics, ed. P. H. Rabinowitz et al., NATO ASI series C 209. D. Reidel, Dordrecht (1987), 177-202. [M5] Mather, J. N., Variantional construction of orbits of twist diffeomorphism. J. Amer. Math. Soc., 4(1991), no. 2, 203-267.

80

Wei Cheng and Chong-Qing Cheng

[MF] Mather, J. N. and Forni, G., Action minimizing orbits in Hamiltonian systems. Transitions to chaos in classical and quantum mechanics, ed. J. Bellissard et a1. Lecture Notes in Math. 1589, Springer-verlag, Berlin, 1991.

[WI] Wang, Q. D., Variational construction of heteroc1inic orbits for the monotone twist maps, in preprint. [W2] Wang, Q. D., More on the heteroc1inic orbits for the monotone twist maps, in preprint.

Multiple solutions and bifurcation of nonhomogeneous semilinear elliptic equations in ~N * Yinbin Deng Department of Mathematics, Huazhong Normal University Wuhan, 430079, P.R.CHINA Yi Li and Xuejin Zhao Department of Mathematics, University of Iowa Iowa City, IA 52242, USA

1

Introduction

In our recent paper [DLZJ, we considered the existence of two solutions for nonhomogeneous problem {

-t::.U+U=/(X,U)+l1-h(X), E HI (]RN) ,

with h (x) E LOO (lRN) f1)

I

xE]RN,

(1.1)1'

U

n H- 1 (]RN)

and 11- > 0. Under the assumptions of

(x, u) E C 1 ((O,OO),]Rl) with respect to u; 1

f2 ) there exist C 1 > 0, C2 E (0,1) such that II (x, t)1 ~ C1 IW- + C2 t for x E ]RN, t E (0,00) and limHoo f(~ ,t) = +00 uniformly for x E ]RN where 2 < p < +00 and N ~ 2; f3) there exists a constant a E (0,1) such that at/: (x,t) ~ all x E ]RN, t E (0, 00) . and h) h(x) E Loo(]RN) n H- 1 (]RN), h(x) ~ 0, h(x) limlzl-too h(x) = 0. We have got the following results:

I (x, t)

~

°

to in ]RN and

"Research supported in part by the Natural Science Foundation of China and NSEC

81

for

Yinbin Deng, Yi Li and Xuejin Zhao

82

Theorem 1.1. If f1 ), f2 ), f3) and h) hold, there exists a positive constant J.L* < +00 such that problem (1.1)J' has at least one minimal positive solution UJ' if J.L E (0, J.L*) and there are no solutions for (1.1)J' if J.L > J.L*; furthermore, UJ' is increasing with respect to J.L E (0, J.L*] and there is a unique solution for (1.1)J'o if p < when N ~ 3.

i/!.2

Defining the variational functional of (1.1) J' by

I(u)=~

{ 2 AtN

where F (x, u)

(lV'uI 2 +U 2 )

= fou f

dx- {

iaN

F(x,u)dx-J.L {

iRN

h(x)udx,

(x, t) dt, we also have the next theorem.

Theorem 1.2. If in addition to fd f3), and f2 ) with p < ir~2 if N ~ 3 we suppose f 4 ) f(x,·) E C 2 (0,+00),

f5) limt-to+ t·

%# =

%# ~ 0 for x E IR N , t ~ 0,

0 uniformly for x E IR N , t

~

0, limt-too t 1 -

q

1%#1 ~ C

uniformly for x E IRN where C > 0 is some constant and 0 < q < N~2' f6) limlxl--+oo f (x, t) for all x E IR N ,

= f (t)

uniformly for bounded t > 0 and f (x, t) ~

f (t)

then problem (1.1)J' has at least two positive solutions uJ" UJ' with UJ' < UJ' if J.L E (O,J.L*) and UJ' is a local minimiser of I(u). In this paper, we will continue to discuss the existence and bifurcation of multiple solutions for problem (1.1)J' for the sub critical case. We will also give a result about the uniqueness of the positive solution of problem (1.1)J' for the critical case and supercritical case. For simplicity, we suppose that f (x, u) = f (u) to be independent of x throughout this paper. More precisely, we will consider the inhomogeneous elliptic problem:

- 6 u + u = f (u) { u E HI (IRN) ,

+ J.Lh (x)

(1.2) J'

under the assumptions: I)

f(t) E Cl( -00, 00) with f'(O) E [0, 1); f(t) is odd and there exists a constant a E (0,1) such that at 2 l' (t) ~ f (t) t ~ 0 for all t E (-00, 00).

II)

f(t) E C2(0,+00), f"(t) ~ 0 for all t ~ 0 and limt-to+tf"(t) = 0, limt-too t 1- q f"(t) ~ C for some constants C > 0 and 0 < q < N"-2 if N ~ 3; 0 < q < 00 if N = 2.

Nonhomogeneous semilinear elliptic equations III)

83

There exists a positive constant B such that f(u) u P- ---t B as u ---t

--1

where 2 IV)

< p < +00 if N

For any given

= 2 and 2

0 we can find Gc > 0 such that (2.4) where 0 < q
0 such that for x E IR N , t > 1.

(2.5)

From (2.4) and (2.5) we deduce that for any c > 0, there exists Gc > 0 such that

If' (t) - f' (0)1 < c + Gct q

for all t > 0

(2.6)

Yinbin Deng, Yi Li and Xuejin Zhao

86 For any fixed R

/" If' (U/L) JIRN

> 0, let BR = {x E ~N Ilxl < R} . We have -

f' (O)llvn - vol 2 dx

(2.7)

: :; r If' (U/L) - f' (O)llvn - vol

2

dx

JBR

+

r

JR,N\BR

If' (u/L)

f' (O)llvn - vol 2 dx

-

: :; r (€ + CouO IVn - vol dx + /" r IV vol dx JIRN 2

JBR

:::; €

JIRN\BR

IVn -

vol 2 dx

2

n -

+ C. [

(€ + Cou~)

(I.. u~+'dx (I.. Iv. - VoI'+' dX) .~, 1 ) '"

+ Co [( /" JIRN\BR

u~+2dX).-h (JIRN\BR r IV

n _

+2dX) qi2].

vol Q

Since Vn --+ Vo strongly in L" (BR) for 2 :::; s < ~~2' {v n } is a bounded sequence in HI (~N). Taking n --+ 00, then R --+ 00 and finally € --+ 0+ we deduce (2 .3). Therefore, Vo achieves >'1. Clearly Iva I also achieves >'1 . Hence we may assume Vo ~ 0 in ~N and Vo satisfies - !:::, Vo

+ (1 - f' (0)) Vo

=

>'1 (i' (U/L) - f' (0)) Vo·

(2.8)

Once again, by the maximum principle for weak solutions we deduce that Vo > 0 in ~N. We will now prove that >'1 > 1. By the definition of u/L we obtain for any J.Ll < J.L2 -!:::,

(U/L2 - u/Lt) + (U/L2 - u/Lt)

=f

(U/L2) - f (u/Lt) + (J.L2 - J.Ll) h (x) (2 .9) ~ f' (u/Lt) (u/L2 - u/Lt) + (J.L2 - J.Ll) h (x).

Multiplying (2.9) by Vo and integrating it over ~N , we get

Nonhomogeneous semilinear elliptic equations

87

By (2 .8) we have

{ 'l (U1'2 iRN

Ul'l)

'lvo

iRN (I' (UI'2) -

= Al {

+ (u M

f' (0))

Ul'l)

-

(U1'2 -

vodx

ul'J vodx +

{ f' (0) (U1'2 iRN

Ul'l)

vodx.

(2.11) By (2.10) and (2.11) we deduce that

iRN (I' (u M ) -

Al {

f' (0)) (U1'2

ul'l)vodx

-

> { (I' (UI'2)

ilR

N

-

f' (0))

(U1'2 - Ul'l)

vodx,

(2.12)

which implies that Al > 1. By the definition of Al we have

Lemma 2.2. Suppose I), II) and h) . Assume that ul' is a solution of (1.2)1' for which Al > 1. Then for any g (x) E H- I (IRN), the problem

has a solution (here we suppose Uo == 0), where Al is the first eigenvalue given by (2.1) .

Proof. Consider the functional

Yinbin Deng, Yi Li and Xuejin Zha(

88

W

E Hi (IR N

From (2.13), Holder's inequality and Young's inequality we havE

).

~ (w) = ~ kN (IV'wI2 + (1 - l' (0)) w 2 ) -~ 2

~~

r (j'(U/L)-1'(0))w ilRN

kN (IV'wI2 +

(1 -

2

dx

dX-!9(X)WdX

l' (0)) w 2 )

dx

r (IV'wI2 + (1 - l' (0))) iRN ~ (~- 2~1) kN IV'wl2 + (1- l' (0))

!

w 2 dx -

__1_ 2Al

By 1) we have 1 - 1'(0)

9 (x) w dx

2

~ IIwll~l - ~E IIgll~-l '

2

~ IIwll~l - ~E IlglI~-l

w dx -

> O. Thus

~ (w) ~ (~ - 2~1) kN IV'wl2 + (1 -

1'(0)) w dx -

(2.15:

> -

[~2 (1 - ~) (1 - 1'(0)) - ~]2 IIwll~l Ai

C

E

2

IIgll~-l

~ -ClIglI~-l if we choose E small. Let {w n } C Hi (I~N) be the minimizing sequence of the variational problerr

From (2.15) we have 1 ( 1) [ 2" 1 - Ai (1 -

2 1, (0)) - 2"E] IIwnllHl::;

CE

2

::; d + T"gIH-l + 0(1)

~ (W n) +

C 2 T"gIH-l

as n - t

E

00.

By Ai > 1 and 1'(0) E (0,1) we deduce that {w n } is bounded in Hi (I1~N) we choose E small. So we may suppose that

wn -tw

weakly in Hi (IRN)

as n - t

00,

wn -tw

a.e. in IRN

as n - t

00.

By Fatou's lemma

j

Nonhomogeneous semilinear elliptic equations

89

We now prove that

as n --t

In fact, by (2.6), for any c

{ II' (uJ»

-

JRN

~ {

JBR(O)

R

I' (O)llwn - WI2 If' (uJ»

+ { JRN\B(O)

~ {

> 0,

-

> 0, we have dx

f' (O)llwn - WI2

If' (uJ»

(c + Gou!)

IW n

-

WI2

dx

{ JRN

dx

+ {

J BR(O)

~c

dx

I' (O)llwn - WI2

-

(2.16)

00.

(c + Gou!) IW n

-

WI2 dx

JRN\BR

IW n

-

WI2

(LR

+ Go

[

+ Go

[( (

dx

U!+2

JRN\B R

dX) on

U!+2

(LR

IW n

-

w1 Q+2 dX) '~21

dX) on ( (

IW n

-

wI Q+2 dX)

q:h].

JRN\BR

Since Wn -t W strongly in U (B R ) for 2 ~ s < ~~2' and {w n } is a bounded lequence in HI (lR.N) , taking n -t 00, then R -t 00, and finally c -t 0+, we :leduce our claim. From (2.16) and the definition of weak convergence we can easily deduce ;hat

IJld

Yinbin Deng, Yi Li and Xuejin Zhc

90 as n -t

00.

Thus

4.> (w)

J

J

= ~ IVwl2 + w2 dx - ~ l' (u/t) w 2 dx -

JIVwl2 + J

< ~ lim n-+oo - 2- lim n-+oo

2

w dx -

J

J

g (x) w dx

~2 n-+oo lim l' (u/t) w~ dx

g (x) Wn dx

= limn-+oo4.> (w n ) = d = inf 4.> (w), wEHl and hence 4.> (w) = d,

which gives that w is a solution of (2.14)11-"

~~2] and let u/t' be solution of (1.2)/t' Then problem (1.2)/t' has its first eigenvalue Al (/1'*) = 1.

Lemma 2.3. Suppose I) - II) and h), Let p E (2,

Proof. Define

by

F (J.L, u)

= .6u - U + f

(u+)

+ J.Lh (x).

Since Al (J.L) > 1 for J.L E (0, J.L*), it follows that Al (J.L*) ~ 1. If Al (J.L*) > 1, th equation Fu (J.L*, u/t") ¢ = has no nontrivial solution. From Lemma 2.2, J maps ~ x HI (~N) onto H-1 (~N) . Applying the implicit function theorem t F we can find a neighborhood (J.L* - 6, J.L* + 6) of J.L* such that (1.2)/t possess{ a solution u/t if J.L E (J.L* - 6, J.L* + 6) . This is contradictory to the definition (

°

J.L* .

3

Propositions and bifurcation

Proposition 3.1. Suppose I)-III) and h). Ifu E HI (~N),

U

> 0, is a solutio

of (1.2)/t' then (i)

U

(x) and

IVu (x)1

have uniform limits zero as

Ixl -t 00;

91

Nonhomogeneous semilinear elliptic equations

(ii) for any c

> 0, there is a constant C > 0 such that u(x)

~

Cexp«-(I+c))Clxl),

Ixl

~

R,

for R > 0 large enough. Proof. We adapt the argument by H. Bresis and T. Kato [BK] to deduce that (IRN) for q large. Letting i > 1, multiplying (1.2)/l by u i and integrating by parts we obtain

u EU

4i (1

+ i)-2 LN

lV'u!(Hi) 12 dx

+ LN u Hi dx = LNf(u)uidX+J.L

I

h(x)uidx . (3.1)

Because of Remark 1.8 and f2 ), we have

{ JRN

f(u)uidx~

{ (Cl UP- l +I'(O)u)ui dx JRN

= Cl

~ Cl ~ C3 LN h(x)uidx

Up-lui dx

+ 1'(0)

{ UHi dx JRN

LN up-Hi dx

+ I' (0)

(c LN (uP-Hi + u

(

JRN

{

JRN

up-Hi dx + C4

{

2

)

dX)

u 2 dx.

JRN

~ C (LN h~ (x) dX) ~ (LN UP+idX) p~; ~ Cl

(LN

h~ (x) dX) + f'(O) LN u p+i dx.

Thus

4i (1 + i)-2 ( lV'u!(1+i) 12 dx JRN

~ ~ ~

{ f (u) u i dx + J.L* { h (x) u i dx (3.2) JRN JRN C { (U P+i +U 2 ) dx+C { h~ (x) dx JaN JRN C { (U p+i +u 2 ) dx+C JaN

92

Yinbin Deng, Yi Li and Xuejin

Let c > 0 be arbitrary. Then for i ~ ir~2

- (p -

" < ct'"+ N"=2 N+2

tp-l+t _

::;

1) we have

+ Cc t N2N- 2

(because ir~2 (p - 1) + i Sobolev's inequality we find

0,

= N}!~i) .

Zh~

for t

~

0

(3 . ~

Z!~) by Young's inequality. Applying th

Again by Young's inequality we havE

Taking c > 0 small enough, we have

( Cc) 1 1- -

2]RN

1

]RN

u q dx::; -C

1

2c RN

u q dx::;

(

U

N2N - 2

Cc)

1- -

-1

2

dx

C

"-

1

2c RN

2N uf'-2 dx ::; C.

Hence u E Lq (lRN) for all q > 0 large. Obviously u satisfies the linear problem - 6 u + u = F (x) =

Choose q > max { !!.f,

ir~2} ·

f (u) + J1,h (x),

By the Holder's inequality in

B2 (x) we get (3.4

Nonhomogeneous semilinear elliptic equations

93

The assumption of Remark 1.8 and f 2 ) yield

IIF (x) ilL

~

(B,(.))

= (/.,(,) [f (u) + ph (x)] I dx ) I

~ (r

(C1u P + f'(O)u

lB 2 (x)

~ (c r

C1u q

+

lB2(x)

~

C

+

(3.5)

JL·h(x))~ dX) ~

f'(O)u~ + (JL·h(x))~ dX) ~

for all q large enough.

It is deduced by elliptic regularity theory that u E C2 •Cl (JRN) . By [GT, Theorem 8.24] we have

IluIIC"(Bl(X» then u (x) ~ 0 as

~ C;

Ixl ~ 00 since u E Lq (JR N ) .

By [GT, Theorem 8.32] ,

lIulb."(Bt(x» ~ C (liuIlC"(B2(X» + IIh (X)IIOB 2(X»)

(3.6)

(3.5)-(3.6) give I(Vu) (x)1 ~ 0 as Ixi ~ 00. Part (ii) can be established as in [S, Proposition 4.4] .

0

Proposition 3.2. Suppose J) - III) and h). Let hex) E CCl(JR N ) n L 2 (JRN). Then U C £CXl(JRN) and U is uniformly bounded in £CXl(JRN), where U is given by {1.3}. Proof. By elliptic regularity theory [GT] we can deduce that U C C 2 .Cl(JRN)n H (JRN). Suppose on the contrary that there is a sequence {Un} C U such that mpxERN Un --+ +00. Take 2

Where a is some constant to be determined later. Clearly, 0 lVn(O) = 1. Because Un are the solutions of (1.2)~, we have 1 -Mn Z-D.wn(x) a ~etting

a

~

= Mn

-D.wn(x)

2

+ Mnwn(x)

= f(Mnwn(x))

~

Wn(X)

+ JLnh(ax + xn) .

~

1 and

(3.8)

we have

+ M~-Pwn(x) = M~-P f(Mnwn(x)) + :r.~l h(ax + xn) .

(3.9)

Yinbin Deng, Yi Li and Xuejin Zh&

94

From 0 ::; wn(x) ::; 1 and the elliptic regularity theory we deduce that wn(x is bounded in C2,(~N). So we can suppose that

wn(x) --+ and hence

Woo

in C2(~N) as n --+

00,

is a nontrivial positive solution of

Woo

-6.w

= Bw p - l

with

lim w(x) 1"'1--+ 00

= 0 and w(O) = 1, [

which is impossible by [CGS], [CL], [GS]. Proposition 3.3. Let u E C 2, (~N) be a solution of

-6.u+u=f(U)+P,h(X) { u E HI (~N) , and let v E HI AI(u)

= inf

(~N)

in~N,

(3 .10

u > 0 in ~N,

be a supersolution of (3.10) . Recall

{IN l'Vvl2 + (1- 1'(0))v2dx Iv E HI(~N), JRN (f'(U) -

2 1'(0))v dx = I}.

(3.11

Then: (i) v 2': u if Al (u) (ii) If Al (u)

> 1; v = u if Al (u)

< I, v (x) 2': u (x), v (x)

= l.

~ u (x) does not hold for all x E ~N.

Proof. For the case Al (u) > 1, if the conclusion were not true, we would hay the set G = {x E ~N, V (x) < u (x)} =/; 0 and meas(G) > O. It is clear fror (3.10) and the fact that v is a supersolution of (3.10) that

- 6. u + (1 - I'(O))u = f (u) - f'(O)U + p,h (x) , - 6. v + (1 - I'(O))v 2': f (v) - I'(O)v + p,h (x). From the convexity of

- 6. (v - u)

f

(3.12

(u) with respect to u we have

+ (1 - 1'(0)) (v - u) 2': f (v) - f (u) - I'(O)(v - u) = [I' (u) - 1'(0)] (v - u) + f" (u + e (v - u)) (v - u)2 2': [I' (u) - 1'(0)] (v - u)

(3.1~

Nonhomogeneous semilinear elliptic equations for some

e (x)

95

E [0,1]. Set w(x) =

{~(V-u)(X)'

x E G, otherwise.

Hence - b,.

w + (1 - f'(O))w

:s;

[I' (u) - I'(O)]w,

which gives

Thus

a contradiction. It is known by Proposition 3.1 that u (x) -+ 0 as Ixl -+ 00, so ..\1 (u) is attained by some 0 in view of Lemma 2.1. Relation (3.13) leads to

f (v - u) [I' (u) - 1'(0)] 0, there exist C, R > 0 such that

U/-,(x) - u/-' (x) ::; C exp (- (1 - 1'(0) - o} Ixl)

for Ixl

~

(3.16)

R,

Proof. From Proposition 3.1 we have

lim (U/-,(x) - u/-,(x)) Ixl-+oo

= O.

1

Let /3 = (1 - 1'(0) - 0) 2" for some 0 > 0 so that 1 - 1'(0) - 0 > O. Set w(x) = U/-,(x) - u/-,(x), then w is the positive solution of

-6w + w = I(w Since w (x) -t 0 as Ixl -t 1 - If (0) - I (w

00,

+ u/-,) - I(u/-,), w E HI(I~N).

(3.17)

by J), II) there exists an R > 0 such that

+ ul') - I (u/-,) - I' (0) w > 1 _ l' (0) w

-

_

~

2

(3.18)

for Ixl ~ R. Let v (x) = mexp (-/3 (Ixl - R)), where m = max{w(x) I {x} = R} > O. For any M > R, set

n (M) = {x En I R < Ixl < M and w (x) Then n (M) is open. For any x E n (M), we have 6 (v - w) (x)

= (/3 2 - /3 (N _ [1 _

1) lxi-I) v(x)

l' (0) _

I (w

::; /3 2 v (x) - [1 - 1'(0) -

= (1- 1'(0) -

= (1 -

> v (x)}.

+ u/-,) - : (0) w -

I (u/-,) ] w

~] w (x)

o)v(x) - (1- 1'(0)

1'(0) - 0) (v (x) - w (x)) -

-~) w(x)

~w (x)

2 ::; (1 - 1'(0) - 0) (v (x) - w (x)) < O.

Nonhomogeneous semilinear elliptic equations

97

By the maximum principle, we obtain, for x E n (M),

v (x) - w (x) ~ min {(v - w) (x) I x E 8n (M)} = min {O, min {(v - w) (x) Ilxl = M}}. Since limlxl-too w (x) = limlxl-too v (x) = 0, this yields, by letting M that

v(x) ~ w(x)

for

Ixl

-T

+00,

~ R,

o

hence (3.16) follows.

Lemma 3.6. Let h(x) E C(JRN) n L 2(JRN) . Then for any g(x) E C(JR N ) n L2(JRN) problem (2.13)/, has a solution w E C2'(JR N ) n H2(JR N ) for all J.L E

(0, J.L*) (again we suppose here Uo == 0). Proof. From Lemma 2.2 we know that (2.13)/, has a solution W E H1(JRN). By the assumptions on h and g, it is known from [[S]; Proposition 4.3] that wE H 2(JRN). The standard elliptic regularity theory yields wE C 2.(JRN). 0 Proof of Theorem 1.3. The conclusion i) comes immediately from Proposition 4.2. As for ii), we define

by

G(J.L, u)

= !::::"u -

u + f( u+) + J.Lh(x) ,

(3.20)

where C2'(JR N )nH2(JR N ) and c(JR N )nL2(JR N ) are endowed with the natural norms. Then they become Banach spaces. It can be verified that F(J.L, u) is differentiable. From Lemma 3.6 we know that for J.L E (0, J.L*),

Gu(J.L, u/,)w

= !::::"w -

w + f~( u/,)w

is an isomorphism of C 2. (JR N ) n H2(JR N ) onto C(JR N ) n L 2(JRN) . It follows from Implicit FUnction Theorem that the solutions of G(J.L, u) = near (J.L, u/,) are given by a continuous curve. Now we are going to prove that (J.L*, u/'.) is a bifurcation point in C 2 . (JRN)n H2(JR N ) by using an idea in [KLO]. To this end, we need the following bifurcation theorem [CR]: Theorem F. Let X, Y be Banach spaces. Let (X, x) E JR x X and let G be a continuously differentiable mapping of an open neighborhood of (X, x)

°

Yinbin Deng, Yi Li and Xuejin Zhao

98

into Y. Let the null-space N(G.,(5.., x)) = span{xo} be one-dimensional and codimR(G x (5.., x)) = 1. Let G>.(5.., x) f/. R(G x (5.., x)). If Z is the complement of span{xo} in X, then the solutions of G(>", x) = G(5., x) near (5.., x) form a curve (>..(s), x(s)) = (5..+,(s), x+sxo+z(s)), where s -+ (,(s), z(s)) E IlhZ is a continuously differentiable function near s = 0 and ,(0) = ,'(0) = z(O) = z'(O) = O. We define G as (3.19), (3.20). We show that at the critical point (J.l*, u,...), the Theorem F applies. Indeed, from Lemma 2.3, problem (2.1) has a solution (PI > 0 in IRN . 1 E C2''''(IR N ) n H2(IRN) if h E C"'(IR N ) n L2(JR.N). Thus G(Ji,* , u,...) = 0, E C 2''''(IR N ) n H2(IRN) has a solution 1 > O. This implies that N(Gu(Ji,*, u,...)) = span{d = 1 is one dimensional and codimR(Gu(Ji,*, u,...)) = 1 by the Fredholm alternative. It remains to check that G,.. (J.l* , u,...) f/. R(Gu(Ji,* , u,.., )) . By contropositive, it would imply the existence of v(x) ~ 0 such that l::,.v - v + f~( u,...)v

= -h(x),

v E C 2''''(IR N ) n H 2(IR N

).

From Gu(J.l*, U,..·)1 = 0 we conclude that JJRN h(X)ldx = O. This is impossible because h(x) ~ 0, h(x) ~ 0 and 1 (x) > 0 in IRN . Applying Theorem F we conclude that (Ji,*, u,...) is the bifurcation point near which, the solutions of (1.2),.. form a curve (J.l* + ,(s), u,.., + S1 + z(s)) with s near s = 0 and ,(0) = ,'(0) = 0, z(O) = z'(O) = O. We claim that ," (0) < 0 which implies that the bifurcation curve turns strictly to the left in (J.l, u) plane. Since J.l = Ji,* + ,(s), u = u,.., + S1 + z(s) in

= 0,

-l::,.u + u - f( u) - J.lh(x)

u > 0, u E C 2''''(IR N ) n H 2(IR N ).

(3 .21)

Differentiate (3.21) in s twice we have -l::,.uss Set here s obtain

+ Uss - 1"( u)u~ - 1'( u)u ss -

= 0 and use that ,'(0) = 0, Us = 1(X)

-l::,.uss

+ Uss -

f~( u,.., )i

= O. and u = u,..,

J.lssh(x)

- f~( u,.., )u •• + ,"(O)h(x)

as s

= O.

= 0 we (3.22)

Multiplying

Gu (J.l*, u,.., )1

=0

by u.s, and (3.22) by 1, integrating and subtracting the results we obtain

which immediately gives ,"(0)

< 0 because f~( u)

~ 0 for all u ~ O.

Nonhomogeneous semilinear elliptic equations

99

Thus

Up, ~ up,' in C 2,Q(JR N ) nH2(JR N ) as p. ~ p.*, Uj.< ~ up," in C 2,Q(JRN ) n H 2(JRN) as p. ~ p.*. Using Lemma 3.6, Proposition 4.2, the implicit function theorem and the uniqueness of positive ground state solution of (1.2)0 we can easily prove that up, ~ 0 in C 2,Q(JRN ) n H2(JR N ) as p. ~ 0, and

o 4

A Uniqueness Result

In this section we shall always assume that hex) satisfies the conditions of Theorem 1.3. We first give a Pohozaev identity. Let (4.1)

G(up"u)

= ioU g(up"s)ds

(4.2)

The following Lemma can be found in [DL1} Lemma 4.1. Ifu E HI (JRN ) is a positive solution of

-!:::.u+u = g(uj. O. Ul.£

(4.7)

On the other hand , by (1.4) we know from [GNN, Li, LNj that

(Vul.£ . x) < O.

It is easy to verify that

t::" 2: 0 for all ul.£ 2: 0, u 2: O. Hence {

88G (Vul.£ . x)dx

J]RN ul.£ This is contradictory to (4.7).

~0

o

Corollary 4.3. Let N 2: 6, p = ~~2' Then (4.3) has no solutions in Hl(JR N) if (1 ·4) holds and J.L is small enough. The Proof of Theorem 1.4: It follows from Theorem 1.1 and Theorem 1.3 that (1.2)1.£ possesses a minimal solution ul.£ if J.L E (0, J.L •• ) for some positive constant J.L ... If (1.2)1.£ has another solution UI.£ and UI.£ 't. ul.£, then UI.£ 2: ul.£ and vl.£ == UI.£ - ul.£ 2: 0 must be a solution of (4.3) . Strong maximum principle implies that vl.£ is a positive solution of (4.3). This is contradictory to Lemma 4.2.

Nonhomogeneous semilinear elliptic equations

101

References [BK]

H.Brezis and T . Kato, Remarks on the Schrodinger operator with singular complex potentials, J. Math . Pures. Appl. 58 (1979), 137-151.

[CD]

Y. B. Deng and D. M. Cao, The Uniquness of Positive Solution for Singular Nonlinear Boundary Value Problems, Sys. Science Math. Sci., 6(1)(1993),25-31.

[CGS] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989) pp. 271-297. [CL]

W.-X. Chen and C.-M. Li, Classification of solutions of some nonlinear elliptic equations Duke Math. J. 63 (1991) pp. 615-622.

[CR]

M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat'l Mech. Anal. 52 (1973) pp. 161-180.

[DL1] Y.B. Deng and Y. Li, Existence and Bifurcation of the Positive Solutions=20 For a Semilinear Equation With Critical Exponent, it J. Diff. Equns. 130(1) (1996) 179-200. [DL2] Y.B. Deng and Y. Li, Existence and of Multiple Positive Solutions For a Semilinear Elliptic Equation Advances DiJJ. Equns. 2(3) (1997),361382. [DLZ] Y.B. Deng, Y. Li, and X.J . Zhao, Multiple solutions for nonhomogeneous semilinear elliptic equations in ]RN, Preprint. [GNN] Gidas, Weimin Ni and L. Nirenberg, Symmetry of the positive solutions of nonlinear elljptic equations in ]RN, Advances in Math. supplementary studies 1 A (1981), 369-402. [GS]

B. Gidas and J. Spruck, Apriori bounds for positive solution of nonlinear elliptic equations, Comm. P. D. E. 6 (1981) pp. 881-901.

[GT]

D. Gilbarg and N.S.Trudinger, " Elliptic Partial Differential Equation of Second Order" 2nd ed., Springer-Verlag, Berlin/Heidelberg/Tokyo/New York (1983).

[KLO] P.L. Korman, Y. Li and T .-C . Ouyang, Exact multiplicity results for boundary value problems with nonlinearities generalizing cubic, Proc. Royal Soc. Edinb., Ser. 126A (1996) pp. 599-616.

102

Yinbin Deng, Yi Li and Xuejin Zhao

[KZ]

M.-K. Kwong and L.-Q . Zhang, Uniqueness of positive solutions of b.u+ f(u) = 0 in an annulus, Diff. Intg'l. Equa. 4 (1991) pp. 583-599.

[Li]

Yi Li, On the positive solutions of Matukuma equation. Duke Math. J. 70 (1993) 575-589.

[LN]

Yi Li and Wei-Ming Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in Rn, Comm. Partial Differential Equations18 (1993) 1043-1054.

[S]

C.A. Stuart, Bifurcation in £P(lRN) for a semilinear elliptic equation, Proc. London. Math. Soc. 57 (1988) pp.511-541.

PROPERTIES OF SOLUTIONS OF SOME FORCED NONLINEAR OSCILLATORS AT RESONANCE C. Fabry and J. Mawhin Department of Mathematics, Universite Catholique de Louvain, B-1348 Louvain-Ia-Neuve, Belgium E-mail :f abry\Oamm. ucl. ac . be, mawhin\Oamm. ucl. ac . be. (Dedicated to Paul H. Rabinowitz for his sixtieth birthday anniversary)

Abstract This article surveys some recent results on the existence of periodic solutions, unbounded solutions, and the boundedness of the set of solutions of bounded nonlinear perturbations of periodically forced resonant linear or asymmetric oscillators.

1. Introduction The concept of resonance occurs in physical systems modeled by a periodically forced linear oscillator. It is present in all parts of physics, astronomy and engineering, and, as mentioned by Feynman [7], 'if we look in [... ] the Physical Reviews, [.. .] every issue has a resonance curve'. If we restrict ourself to undamped oscillators, the physical phenomenon of resonance consists in observing oscillations with increasing amplitude when the oscillator is submitted to a sinusoidal external force whose period is equal to the natural period of the oscillator. Assume more generally that the phenomenon is modeled by the linear equation u"(t)

+ w~u(t) = e(t),

(1.1)

where Wo > 0, and e is a continuous periodic function whose period is normalized to 211'. Thus e hast the Fourier series 00

e(t) '" ao

+L

(an cos nt + bn sin nt),

n=l

with ao

= -1 121T e(t) dt, 211'

0

an

= -1 121T e(t) cos nt dt, 11'

0

bn

= -1 121T e(t) sin nt dt. 11'

0

An elementary mathematical analysis shows that Eq. (1.1) has all its solutions unbounded if and only if Wo = n for some integer n ~ 1 and a;' + b;' 1:- o. 103

c.

104

Fabry and J. Mawhin

When Wo = n, a positive integer, and an = bn = 0, all the solutions of Eq. (1.1) are 27T-periodic. When Wo is not an integer, Eq. (1.1) has a (unique) 27Tperiodic solution for each forcing e, and all its solutions are bounded (indeed quasi-periodic) over lR. Consequently, for the concrete system modeled by Eq. (1.1), physical resonance (i.e. the unavoidable presence of oscillations with increasing amplitude) occurs if and only if Eq. (1.1) admits no 27T-periodic solution. For a forced nonlinear oscillator modeled by the equation u"(t)

+ h(u(t))

= e(t),

(1.2)

the above dichotomy between the existence of unbounded and of periodic solutions does not hold in general. A large literature deals with finding conditions upon hand e for which Eq. (1.2) has at least one 27T-periodic solution. Those results have been essentially proved using some fixed point argument for the corresponding Poincare's operator in ]R2 or an associated nonlinear operator in a space of 27T-periodic functions. Much less work has been devoted to the existence of unbounded solutions, using a variety of approaches, and to proving the boundedness of all solutions. Few results deal with the relation between the existence of periodic and unbounded solutions. The pioneering result in the last category is Massera's theorem [17], insuring the existence of a 27T-periodic solution of Eq. (1.2) if the corresponding Cauchy problem is uniquely and globally solvable, and if Eq. (1.2) has one solution bounded in the future or in the past. Consequently, under the appropriate hypotheses for the Cauchy problem, the absence of 27T-periodic solutions for Eq. (1.2) implies that all its solutions are unbounded in the past and in the future. But, as we shall see, the converse of this result is not true in general for nonlinear oscillators. The aim of this paper is to survey some recent results on the qualitative structure of the solutions of Eq. (1.2), when the nonlinearity h belongs to various classes of mappings like some bounded perturbations of n 2 u, the asymmetric (or jumping) nonlinearities, and some bounded nonlinear perturbations of asymmetric oscillators. In view of the previous discussion of the linear case, special attention will be given to the existence of 27T-periodic solutions, of unbounded solutions and to the bounded ness of the set of all solutions of the equation.

Solutions of forced nonlinear oscillators at resonance

105

2. The resonant linear oscillator Let us consider the forced linear oscillator u"(t)

+ n 2 u(t)

= e(t),

(2.1)

where n is a nonnegative integer and e a 27r-periodic continuous function . The discussion in the Introduction can be formalized as follows: 1. Eq. {2.1} has a 27r-periodic solution if and only if

121r e(t)cosntdt = 0

and

121f e(t)sinntdt = O.

(2.2)

2. Eq. {2.1} has all its solutions unbounded if and only if

121r e(t) cos nt dt ::j:. 0

ar

121r e(t)sinntdt::j:. O.

(2.3)

The first statement is more akin to the mathematical aspect of the resonance problem (namely the one which has a corresponding formulation for other boundary conditions than the periodic ones), and the second one to its physical characterization. Indeed, from a physicist or engineer's viewpoint, condition (2.2) is a way to avoid the (generally catastrophic) apparition of unbounded oscillations.

If we define the real ~ anti-periodic (and hence IR -+ IR by

1 Se(fJ) := 27r

2: -periodic) function Se :

121r e(t) sin n(t + 0) dt

(2.4)

0

= (2~ 121r e(t) cosntdt) sin nO + (2~ 121r e(t) sin ntdt) cos nO,

(0 E IR),

then the above result can be stated in the following form, which is particularly well suited for the comparison with the subsequent results for the nonlinear case.

Theorem 2.1 Let n

~

1 be an integer.

1. Eq. {2.1} has a 27r-periodic solution if and only if Se == O. 2. Eq. {2.1} has all its solutions unbounded if and only if the function Se takes both positive and negative values. Notice that, because of the special nature of the function Se, the two behaviors described in Theorem 2.1 are the only possible ones and exclude each other.

C. Fabry and J. Mawhin

106

3. The boundedly perturbed linear resonant oscillator Since the pioneering work of Lazer and Leach [11] in 1969, a large literature has been devoted to the study of bounded perturbations of linear oscillators at resonance, namely

u"(t)

+ n 2 u(t) =

f(u(t))

+ e(t),

(3.1)

where n is a positive integer, e is continuous and 271"-periodic, and f : IR -+ IR. is continuous and bounded. Most ofthe results are related to extending to Eq. (3.1) the first formulation of the resonance property for Eq. (2.1). We shall restrict, for simplicity, to the case where

f(-oo):=

lim f(u),

u ..... -oo

and

f(+oo):=

lim f(u),

u ..... +oo

exist, although more general situations can be handled. One way of formulating the Lazer-Leach's result (Theorem 1.1 of [11]) goes as follows.

Theorem 3.2 Eg. (3.1) has a 271"-periodic solution if the function 1

- [J(+oo) - f(-oo)] + Se(O), 71"

(3.2)

(0 E IR),

has a constant sign (i.e. is positive for all 0 E IR or negative for all 0 E IR). The original proof of Lazer and Leach was based upon a clever application of Schauder's fixed point theorem, which required a lot of technicalities. Simpler proofs use topological degree arguments (see e.g. [10]). The sufficient condition (3.2) was written in [11] in the form

!en 1< .!.If(+oo) - f(-oo)l, 71"

(3.3)

where

en =

1

271"

1211" 0 e(t) exp( -int) dt,

so that

12,.1=

[( 11211" e(t)cosntdt) 2+ (1 271"

0

r211"

271"Jo

e(t)sinntdt

) 2]1/2

Solutions of forced nonlinear oscillators at resonance

107

This condition is necessary in the class of functions f such that

If(+oo) - f(-oo)1 > 0 and

min{J(-oo) , f(+oo)}:S f(u):s max{J(-oo),f(+oo)},

(u E JR).

For example, the equation u"(t)

+ n 2 u(t) = arctan u(t) + e(t)

(3.4)

has a 27r-periodic solution if and only if condition (3.5)

holds. Theorem 3.2 can be seen as a (purely nonlinear) extension of the first assertion in Theorem 2.1. It is less known that Lazer and Leach also proved a (weaker) extension to Eq. (3.1) of the second assertion in Theorem 2.1 (Theorem 1.2 of [11)). Here, f does not need to have limits at ±oo. Theorem 3.3 Eq. (3.1) has no 27r-periodic solution if the condition

lenl ~ .!.7r

[sup IR

f - inf f] > O. IR

(3.6)

holds.

The proof, by contradiction, is based upon delicate ad hoc arguments on integral identities deduced from the assumption of the existence of a 27r-periodic solution. When f is such that the Cauchy problem for Eq. (3.1) is uniquely solvable, it follows from Massera's theorem mentioned above [17] that Theorem 3.3 is equivalent to the following assertion, which can be seen as a nonlinear counterpart of the second statement of Theorem 2.1. . Theorem 3.4 Eq. (3.1) has all its solutions unbounded in ] - 00,0] and in [0, +oo[ if condition (3.6) holds.

C. Fabry and J. Mawhin

108

In [1], Alonso and Ortega have generalized Theorem 3.4 in various ways. They use the following result, whose proof is quite straightforward. Let X be a Banach space, F : X -+ X and let V : X -+ ]R be bounded above on bounded sets. Lemma 3.5 Assume that there exist V(F(~))

2:

V(~)

r >0

+r

and p I~I

if

> 0 such that

2:

(3.7)

p.

> sUPI(I~p V(~), the sequence of iterates defined

Then, for each ~o with V(~o) by

~n+1 = F(~n),

(n = 0,1, ... )

satisfies

lim I~nl

n-t+oo

= +00.

(3.8)

If condition {3.7} holds for all ~ E X, then condition {3.8} holds for all ~o E X. If X has finite dimension and F is continuous, condition {3.7} on X can be replaced by V(F(~)) > V(~) for all ~ E X .

Let the function f in Eq. (3.1) be bounded and such that the Cauchy problem for Eq. (3.1) is uniquely solvable. The first application of Lemma 3.5 to Eq. (3.1) refines the conclusion of Theorem 3.4. Theorem 3.6 If condition {3.6} holds, then every solution of Eq. {3.1} satisfies the relation lim [u(t)2 + U'(t)2] = +00. Itl-too

The proof consists in applying Lemma 3.5 with X = operator associated to Eq. (3.1)

F(6,6)

= [u(211"j6,6),u'(211",6,6)]

with

V(6, 6) and if> such that

]R2,

= 6 sin if> -

n6 cos if>,

to the Poincare's

Solutions of forced nonlinear oscillators at resonance

109

For example, all the solutions of Eq. (3.4) are unbounded when

(3.9) Theorem 3.6 was motivated by [11) and by a more recent partial result of Seifert [22). Using fine tools of topology in the plane, Seifert had proved that all solutions of Eq. (3.1) are unbounded on [0, +oo[ when condition

!en I 2:: [s~p I - i~f I] > 0, a stronger assumption and a weaker conclusion than in Theorem 3.6. In 1966, Littlewood raised the question of the boundedness (over JR) of all the solutions of a nonlinear oscillator u"(t)

+ h(u(t))

= e(t),

(3.10)

where h is such that hex) --t ±oo when x --t ±oo, and e is continuous and 27r-periodic, and gave examples of superlinear functions h (Le. h(x)/x --t +00 when Ixl --t 00), for which Eq. (3.10) admits unbounded solutions [13,14). Subsequently, large classes of superlinear functions h for which Littlewood's problem has a positive answer have been given, as well as further examples for which it has a negative answer. References can be found in (19), where Ortega has initiated the study of Littlewood's problem for equations of the form (3.1). In the special case where I is the piecewise linear function defined as follows (for some fixed L > 0) -L

I(x) =

{

il

-Lx il L

il

x 2:: 1 Ixl:::; 1 x:::; -1,

Ortega has proved that, if e is of class C S , and the function (3.2) is of constant sign, which is here equivalent to the condition ~ I 2L and 8 E [0,211" /n[ (see e.g. (9)). Dancer has introduced in [3] the following extension of the function 8 e (8) defined by relation (2.4)

°

1

1;e(8) := 211"

10r

27r

e(t)O"(t + 8) dt,

(8 E JR),

(4.5)

2:

(a -periodic function), and extended to Eq. (4.1) the first assertion of Theorem 2.1. Theorem 4.9 If the function 1;e is of constant sign on JR, then Eq. (4.1) has

a 211"-periodic solution. Dancer's proof is based upon a combination of perturbation and degree arguments. This result has been completed as follows by Fabry and Fonda [5]. Theorem 4.10 If the function 1;e has more than two zeros in [0,211"[, all simple, then Eq. (4.1) has a 211"-periodic solution.

The proof of Theorem 4.10 is based upon degree arguments. Using the change of variables (u, u') --+ (p,8) defined by

u(t) = p(t)O"[8(t)],

u'(t) = p(t)O"'[O(t)],

the authors show, by a contradiction argument, that the set of all possible 211"-periodic solutions of the family of equations

ul/(t)

+ /1-u+(t) -

IIU-(t) = ).e(t),

oX E rO,ll,

Solutions of forced nonlinear oscillators at resonance

113

is a priori bounded independently of >., and that the associated coincidence degree is equal to 1 - z, if 2z denotes the number of zeros of ~e in [0,211" /n[. The following result is an extension to Eq. (4.1) of the second assertion of Theorem 2.1. Theorem 4.11 If the function ~e takes both positive and negative values and if all its zeros are simple, then there exists R > 0 such that every solution u(t) of Eq. (4-1) with for some to E JR, is such that

as t

-t

+00 or t

-t

-00.

The proof of this result in [2] is distinct from the one in [1], and is obtained in applying to the associated Poincare's map some results of the authors on the dynamics of a class of mappings on the plane which have, in polar coordinates, an asymptotic expansion of the form

(h = O+211"E+~J.tl(O)+o(r-l) r

q

rl

=

r

+ J.t2(O) + 0(1),

valid when r -t +00. In this expression, ~ is a rational number, J.tl and J.t2 are continuous, 211"-periodic, and satisfy conditions which will not be made explicit here. Notice that Alonso and Ortega have proved a more general result in [2]. They consider the case where J.t > 0 and v > 0 are such that 1

1

- + - E Q. .,ffi .jV Through the introduction of a suitable extension of the function ~e, they show ~hat, for any such couple (J.t, v), there exists functions e for which large solutions )f Eq. (4.1) are unbounded. Among the functions for which unbounded solutions exist, the case of (11, where (1 is defined by (4.4), deserves a particular interest. Indeed, for a fixed value of lIellL2, the choice e = (1' will maximize II~~ IlL'' >. This, in turn, will naximize the mean asymptotic rate of growth of u 2(t)+u /2 (t). More precisely, it :an be shown that, with e = (11, and provided that u(tO)2+u' (tO)2 is sufficiently

C. Fabry and J . Mawhin

114

large, the average over a period of [U /2 (t) + J.t(U+(t))2 + v(U-(t))2j1/2 will grow, asymptotically, with a rate equal to 1I1:~,liLoo = 1/2. Notice that, when e = a', Eq. (4.1) admits the particular solution ~ta(t), for t 2: O. We do not know however if all solutions are unbounded. One could think that for Eq. (4.1), like in the linear case, the existence of a 27r-periodic solution should imply the boundedness of all solutions. In (2), Alonso and Ortega have shown that it is not the case, by exhibiting 27r-periodic functions e(t) such that the corresponding Eq. (4.1) admits a 27r-periodic solutions, but all its solutions with large initial conditions are unbounded. This is the case for the equation

u"(t)

+ J.tu+(t)

- vu-(t) = cosrt,

(4.6)

when J.t f v satisfy condition (4.3) for some n 2: 1, and r 2: 1 is an integer. It is shown in [2] that there exist infinitely many r for which the corresponding function 1:cos r. ((') has the form Cr sin rO with Cr f 0, and hence satisfies simultaneously the assumptions of Theorems 4.10 and 4.11. Thus there is coexistence of periodic and unbounded solutions. Concerning Littlewood's problem, Ortega has shown in [18] that all solutions of Eq. (4.1) are bounded over IR when e(t) = 1 + b(t), with b of class C 4 and its C 4 norm sufficiently small. Very recently, Liu Bin [16] has proved, using Ortega's theorem [19] for planar twist mappings, the following result. Theorem 4.12 If e E C6 and the function 1:e(O) is of constant sign, then all solutions of Eq. {4.1} satisfy condition {3.11}. We refer to [20,21] for very nice informal discussions of those results and of related questions , which are treated in more details in [16,18,19] .

5.The boundedly perturbed resonant asymmetric oscillator In this section, we describe some recent results of the authors [6] concerning the periodic and the unbounded solutions of nonlinear equations of the form

u"(t)

+ J.tu+(t) - vu-(t) = f(u(t)) + g(u(t)) + e(t),

(5.1)

where J.t and 1/ are positive numbers satisfying condition (4.3) for some integer n 2: 1, e is continuous and 27r-periodic, the function f is locally Lipschitzian and

Solutions of forced nonlinear oscillators at resonance

115

3.dmits limits f(±oo) at ±oo, and the function 9 is bounded, locally Lipschitzian 3.lld has a sub linear primitive. The following extension of Theorems 4.9 and 4.10 can be obtained by adapting the proof of [5] to the more general setting. Theorem 5.13 If the junction (5.2)

has a constant sign or has more than two zeros in [0,211"[, and all its zeros are simple, then Eq. (5.1) has a 211"-periodic solution. Again, if 2z denotes the number of zeros of the function (5.2), then the topological degree associated to the periodic problem for Eq. (5.1) is equal to 1- z. Theorem 4.11 is extended as follows. Theorem 5.14 If the junction (5.2) has zeros, all being simple, then any solution of Eq. (5.1), written as

x(t) = p(t)lP(t + B(t)), x'(t) = p(t)lP'(t + B(t)), with p(O) sufficiently large, is unbounded either in the past or in the juture. The proof of Theorem 5.14 uses quite different techniques than the ones :lescribed in the previous sections. Writing the solutions of Eq. (5.1) under the :orm

x(t)

= p(t)lP(t + B(t)),x'(t) = p(t)lP'(t + B(t)),

IVhere lP is defined in Eq. (4.4), and denoting by ~(B), for the sake of brevity, ;he function defined by Eq. (5.2), the basic ingredient is a lemma showing that, or large solutions, p(t)~(B(t)) remains close to a constant on a time scale of ;he order of p(O). More precisely, ~emma

> 0, T > 0, there exists R > 0 such that for p(O)

~

~(B(O))I p(t) - p(O) ~(B(t)) ~ TJP(O)

(5.3)

5.15 Given TJ

I or t E [0, Tp(O)].

R,

c. Fabry and J. Mawhin

116

This lemma is proved through a delicate application of Riemann-Lebesgue type theorems for the asymptotic behavior of oscillatory integrals and of the classical averaging method [23]. Coming back to the proof of Theorem 5.14, as

0 and 51 -equivariant -+ (V, V - {O}) such that

gradient

1. there is an 51 -equivariant gradient homotopy '\l H t such that (a) '\lHt : (cl(D",(V»,S",(V» (b) '\l Ho

= '\l I

and '\l HI

-+ (V, V - {O})

= '\l 10,

map

Variational SI-symmetric problems with resonance at infinity

(c) \7 H t- l (O)

n cl(Do:(V)) =

123

{O} lor any t E [0,1]'

2. there is an SI-equivariant gradient map \7cpo : (cl(DO:(VI0)), 0) -+ (VI0, 0) such that ilv = (Vl,V2) E V = V?EDV2o then \7fo(v) = \7fO(Vl,V2) = (\7cpo(vd, \7 2 f(0)(V2)) . See [5] for an appropriate homotopy. The following formula for topological degree is well known deg((f,g),fh x O2 ,(0,0)) = deg(f,OI'O)· deg(g,02'0), where (f, g) : (0 1 x O2 ,8(0 1 x O2 )) -+ (~n X ~m, ~n X ~m - {(O, On). The following theorem is an analogue of the above mentioned property in the case of SI--equivariant gradient maps and degree for SI--equivariant gradient product maps. Let us introduce a structure of ring in the group

Z (~Z) ED

defining

the multiplicative structure in the following way:

0: * /3 for 0:

= (0:0 . /30, 0:0 . /31 + /30 . 0:1, 0:0 . /32 + /30 . 0:2, ... )

= (0:0,0:1,0:2, ... ), /3 = (/30, /31, /32, .. .) E

Z (~Z). ED

From now on the

above defined ring will be denoted U(SI) and called tom Dieck ring, see [8] for definition of the tom Dieck ring for any compact Lie group. Theorem 2.3 (Cartesian product formula) Let 0 1 C V and O2 C W be open, bounded and SI-invariant subsets. Let \7 f : (0 1 ,80 1 ) -+ (V, V - {O}) and \7g : (0 2 ,80 2 ) -+ (W, W - {O}) be SI-equivariant gradient maps. Then, DEG((\7f, \7g), 0 1 X O2 ) = DEG(\7 f,Od *DEG(\7g, O2). See [13] for a proof of this theorem. The following lemma is an analogue of Splitting lemma at the origin but we deal here with infinity instead of the origin. This lemma yields information about behaviour (up to SI--equivariant gradient homotopies) of asymptotically linear SI--equivariant gradient map outside of a sufficiently large disc centred at the origin. From now on we put V = VtO ED \1200, where Vl°O = ker \7 2 f(oo) and V2°o = im \7 2 f(oo) . Lemma 2.4 (Splitting lemma at infinity) Denote by \l I and SI-equivariant gradient C l -map such that

1. \7f-l(O) is bounded, 2. \7f(v)

= \7 2 f(00)(v) + \71J(v),

3. \7 2 f(oo) is degenerate, symmetric matrix,

: (V, 0) -+ (V, 0)

N. Hirano and S. Rybicki

124

4. I \7 27](v) 1-+ 0, as I v 1-+ 00. Then, there are numbers (31, (32 > 0 and Sl-equivariant gradient map

such that 1. there is an Sl-equivariant gradient homotopy \7 H t such that (a) \7Ht : (cl(D.a,(Vt:l) x D.a2(V2oo)),8(cl(D.a, (V!"O) x D.a2(V2oo))))-+ (V,v - {O}), (b) \7Ho

= \7f

and \7Hl

= \7foo, -+

2. there is an Sl -equivariant gradient map \7 tpoo : V100

\7 foo (v) = \7 foo( VI, V2) V = V100 EEl V200 •

=

V100 such that (\7tpoo (VI), \7 f( 00)(V2)) for V (VI, V2) E 2

=

In order to prove Splitting lemma at infinity we will need the following three technical lemmas.

Lemma 2.5 Denote by \7 f : (V,O) map such that 1. \7 f(v)

-+

(V,O) an Sl-equivariant gradient C 1 _

= \7 2 f(oo)(v) + \77](v),

2. \7 2 f(oo) is degenerate, symmetric matrix, 3.

I \7 27](V) 1-+ 0,

as I V 1-+

00 .

Fix (3 > 0 and Sl-equivariant map 'I/J E CO(cl(D.a(Vloo)) x [0,1], \1200). Define a map f* : cl(D.aCVioo)) x V200 x [0,1] -+ V200 as follows f*(VI, V2, t) = 7r(\7f(Vl,V2 +'I/J(VI,t))), where 7r: V -+ V200 is the orthogonal, Sl-equivariant projection. Then. there is (3~ > 0 such that if f* (VI, V2, t) = 0, then I V2 I< (3~. Proof. For abbreviation, we let A stand for \7 2 f(00)W2"'" that

f*(VI, V2, t)

= =

First of all notice

A(V2 + 'I/J(Vl, t)) + 7r(\77](Vl' V2 + 'I/J(Vl, t))) A(V2) + A ('I/J (VI ,t)) + 7r(\77](Vl, V2 + 'I/J(Vl, t)))

and that

I (Vl,V2 +'I/J(Vl,t)) I = I (Vl,V2) + (O,'I/J(Vl,t) I ~ I (Vl,V2) I-I (O,'I/J(Vl,t) I ~ I (Vl,V2) I-m ~ I V2 I - I VI I - m ~ I V2 I - (3 - m,

(2.1)

Variational Sl-symmetric problems with resonance at infinity where m

=

125

I 'ljJ(Vl,t) I.

sup (til ,t)Ecl(Dp(Vt"» x [0 ,1)

Summing up, if (VI, t) E cl(Dp(Vl00)) X [0,1) and I (VI, V2, t) I~ 00, then I (Vl,V2 +'ljJ(Vl,t)) I~ 00. From the assumptions it follows that

c > 0 thereis R > 0 s.t. I V I ~ R then I V'1](v) I < c· I V I . (2.2) Fix c and choose R as in (2.2) and notice that by (2.1) we have: if (VI , V2, t) E cl(Dp(Vl00)) X V2 X [0,1) and I V2 I> R + m + {3, then I (Vl,V2 +'ljJ(Vl , t)) I> R, and consequently by (2.2) forany

°O

17l"(V'1](Vl,V2 + 'ljJ(Vb t))) I ~ I V'1](Vl,V2 +'ljJ(Vl , t)) 1< c·1 (Vl,V2 +'ljJ(Vl , t)) I· Fix c < IA;ll-l, choose R as in (2.2) and (VI , V2, t) E cl(Dp(Vl00)) such that I V2 I > R + m + (3 and notice that

X

V2°O

X

[0,1)

I f*(Vl,v2,t) I ~ I A(V2) I-I A('ljJ(Vl,t)) 1-17l"(V'1](Vl,V2 +'ljJ(Vl,t))) I ~ I (A)-l 1-1 . I V2 I - I A I . m - I V'1](Vl' V2 + 'ljJ( VI, t)) I ~ I A-I 1-1 . I v2 I I A I . m - c· I (VI, V2 + 'ljJ( VI, t)) I~ I A-I 1-1 . I V2 I - I A I . m -1 ~ · 1(Vl,V2 +'ljJ(Vl,t)) I ~ I A-I 1-1 . 1v21-1 A I·mIA;ll-l . (I V2 I + I (Vl,'ljJ(Vl,t)) I ~ IA-;1-1·1 V2 I -I A I · mIA;11-1. 1(Vl,'ljJ(Vl , t)) I~ IA-;r1·lv21_IAI·m_ IA;ll-l . ({3+m). -1

Taking into consideration the above to complete this proof it is enough to take > max{R+m+{3, 2·1 A-ll · 1A l·m+m+{3}.

{3~ such that {3~

Lemma 2.6 Denote by map such that

o

V' f : (V, 0)

1. V'f(v) = V'2f(00)(v)

~ (V,O)

an Sl-equivariant gradient C 1-

+ V'1](v),

2. V'2 f (00) is degenerate, symmetric matrix, I V'21](v) I~ 0 as I V I~ 00. Fix (3 > 0 and a map 'ljJ 3.

E CO (cl(Dp(Vl00)) , V2OO) . Define a map V2°O by the formula f* (VI, V2, t) = t . 7l"(V' f (Vl ' t . V2 + 'ljJ( vt})) + (1 - t 2) . V'2 f (00)(V2),

f* : cl(Dp(Vt')) x V2°O x [0, 1)

where

7l" : V

~

~

V2°o is an orthogonal, Sl-equivariant projection. Then, there is then I V2 1< {3i·

(3i > 0 such that if f*(Vl,V2,t) = 0,

126

N. Hirano and S . Rybicki

Proof. For abbreviation, we let A stand for V 21(00) 1V2 that

°o.

and that for any

M =

First of all notice

> 0 there are K€ > 0 and {3€ > 0 such that for any {30 > (3,

€

I V7J(V1,t·V2+1/J(Vt}) 1< €·({3+{3o+m)+K€,

sup (v,t)Ecl(DIl (VlOO)) x Silo (V2OO) x [0,1]

where m =

11/J(vt}

sup VI

I.

Fix

€


max{2· I A-I I ·(1 A I . m + € . ({3 + m) + K€),{3€}. Hence for (VI, V2, t) E cl(Dp(V{lO)) X 8po (V200 ) x [0,1) we have

I f*(Vl,V2,t) I ~ I A-I 1-1 . I v21 -I A I . 11/J(V1) II V7J(V1, t· V2 + 1/J(V1)) I ~ I A-I 1-1 . {30- I A I . m€ .

({3+{3o+m)-K€ ~ IA-;1-

1 •

{3o-IAI · m-€· (m+{3)-K€>O.

Taking into consideration the above it is clear that in order to complete this proof it is enough to take {3~ > {30 .

o Lemma 2.7 (Implicit function theorem at qui variant C2 -map h = 7r 0 VI: VI ffi V2 -+ orthogonal, 8 1 -equivariant projection and VI: C l -map. Assume that

infinity) Consider an 8 1 -eV2, where 7r : V -+ V2 is an V -+ V is an 8 1 -equivariant

2. V 2 /(00) is nondegenerate, symmetric matrix, 3.

I V 27J(v) 1-+ 0,

as I V 1-+ 00.

number (30 > w* : VI - cl(D po (Vl)) -+ V2 such that Then,

there

is

o

and

8 1 -invariant

C 1 -map

A proof of the above lemma (nonequivariant case) one can find in [2). The above equivariant version is an easy consequence of a nonequivariant one. We are now in a position to prove the Splitting lemma at infinity.

Variational SI-symmetric problems with resonance at infinity

127

Proof of Splitting lemma at infinity. Let 71" : V -+ V2°O be an SI-equivariant orthogonal projection. Consider SI-equivariant maps It = (I d - 71") 0 'V 1 : V -+ Vl°O and h = 71" 0 'VI: V -+ V2°O . Let us apply Lemma 2.7 to map h. For v = (VI, V2) E V = Vl oo EB V2°O we have h(VI,V2) = 'V 2/(oo)(V2) + 7I"('V1J(VI,V2)) . Notice that the above defined map h satisfies all the assumptions of Lemma 2.7. Applying Lemma 2.7 we obtain (30 > 0 and SI-equivariant CI-map w· : Vloo - cl(D,Bo(VIOO)) -+ V2°O such that h(Vl,V2) = 0, VI E Vloo - cl(D,Bo(VIOO)) iff V2 = w·(vt}. From assumption 1. it follows that without loss of generality one can assume that

(I - 71")( {v E V : 'V I(v) = O})

c cl(D,Bo (Vl00 ))

(2.4)

Let w : Vl°O -+ V2°O be any SI-equivariant C l -extension of W· . Fix any (31 > (30 and notice that 'V 1(VI , V2) = 0 and I VI I2:: (31 iff II (VI, W(vt}) = O. From (2.4) we conclude that (2 .5) Put (3 = (31 and 1/J(Vl, t) = t . w(vd in Lemma 2.5 . Applying Lemma 2.5 we obtain (3~ such that 'V/- l (O) C cl(D,Bl(Vl00 )) x cl(D,B~(V200)). Define two families of SI-equivariant functions hi, h~ : VlOO EB V2°O -+ IR[l, 0) by the formula

and

t E [0,1). Consider SI-equivariant gradient homotopy

'Vhi : cl(D,Bl(Vl00 )) x cl(D,B~(V200)) -+ V,t E [0,1). 00

What is left is to show that homotopy 'Vhi does not vanish on 8(cl(D,Bl (Vl ) x D,B2 (V200 ))) = S,Bl (Vl00 ) x cl(D,B~ (V200 )) U cl(D,Bl (VlOO)) x S,B~ (V200 ). Notice that by the chain formula we obtain 1. 'VhHvI, V2)

=

2. 'VhHvl,V2)

= 0 iff

(It (VI, V2+t 'W(VI))+ h(Vl, V2+t·W(VI))O(t·Dw(VI)), h(VI, V2+ t 'W(Vl))),

+ t· w(vt}) + h(VI,V2 + t· W(Vl)) 0 (t · DW(Vl)) = 0, (b) h(Vl, V2 + t· w(vt}) = O. (a) It(Vl,V2

N. Hirano and S. Rybicki

128

Fix (VI,V2,t) E S.BI(Vt') x cl(D.BI(Vt')) x [0,1] and notice that 2 t h(VI, V2+t·w(vd) = iffv2 = (I-t)·w(vd . And consequently, V'h l (VI,V2) = iff V'hHvI, (1 - t) . w(vd) = iff h(VI,W(VI) = 0. In view of (2.5) the last equality is impossible. Fix (VI,v2,t) E cl(D.BI(VIOO)) x S.B~(V200) x [0,1] and notice that Lemma 2.5 yields that h (VI, V2 + t . W( VI)) ::fi 0. W have just shown that homotopy V'hi is well-defined SI--equivariant gradient homotopy. Put 'IjJ(vd = w(vd and {3 = {3I in Lemma 2.6. Applying Lemma 2.6 we obtain number {3~. Consider SI--equivariant gradient homotopy V'h~ : cl(D.BI (VIOO)) x cl(D.B~(V200)) -+ V, t E [0,1] and notice that

°

1

°

°

V'h~(VI' V2) = (h(VI, t · V2 + w(vd) + h(VI, t · V2 + w(vd)o . DW(VI)' t · h(VI, t· V2 + w(vd)) + (1- t 2) . (0, V'2 f(OO)(V2))

+ w(vd) + h(VI, t· V2 + w(vd) 0 Dw(vd t· h(VI, t· V2 + W(VI)) + (1- t 2) . V'2 f(OO)(V2) = 0.

(a) h(VI, t · V2 (b)

= 0,

Fix (VI, V2, t) E S.BI (VIOO) X cl(D.B~(V200)) x [0,1]. Applying Banach fixed point theorem to the second coordinate of homotopy V' h2 one can show that the second coordinate is equal to iff V2 = 0. If V2 = then the first coordinate is equal to h(VI,W(vd) . From (2.5) it follows that h(VI,W(vd)::fi 0. Applying Lemma 2.6 we show that the second coordinate of the homotopy V'h~ does not vanish on cl(D.BI(VIOO)) x 8(D.B~(V200)). W have just shown that homotopy

°

°

is well-defined SI--equivariant gradient homotopy. In order to complete our proof it is enough to take {32 = max{{3J, {3n and define an SI--equivariant gradient homotopy

as follows for t E [0, ~] for t E [~, 1] Homotopy V'Ht is a required homotopy. Of course CPoo(vd

= h(VI,W(vd) . o

129

Variational Sl-symmetric problems with resonance at infinity

3

Asymptotically Quadratic Functions Degenerate at Infinity

In this section we prove the existence of nontrivial zeros of asymptotically linear Sl--equivariant gradient maps. It is worth pointing out that in theorems of this section the derivative at infinity, V 2 f(oo), is degenerate. The following class of maps will be considered throughout this section. Definition 3.1 An Sl-equivariant gradient Cl-map V f : (V,O) said to be

--+

(V,O) is

1. asymptotically linear at the origin, if (a) V f(v) =

v

(b)

2

v 2 f(O)(v) + V1]o(v),

f(O) is a symmetric matrix,

(c) V1]o(v) = 0(1 v I), at v = 0, The class of asymptotically linear maps at the origin will be denoted by Mo(V) · 2. asymptotically linear at infinity, if

,,2

= f(oo)(v) + V1]oo(v), 2 (b) v f (00) is a symmetric matrix, (c) I V 21]oo(v) 1--+ 0 as I v 1--+ 00.

(a) "f(v)

The class of asymptotically linear maps at infinity will be denoted by ACoo(V). Finally put AC:'" (V)

= AC, (V) n ACoo (V) .

We are now in a position to formulate and prove the first theorem of this section. We give the complete proof of this theorem. Whereas in the proofs of other theorems of this section we will refer the reader to the proof of the first one. Theorem 3.2 Let Vf E AC:"'(V). Assume that

1.

l VloS

= {O} n

imd

ooSl Vl

= {O}, nl

n2

2. V ~ $1R[ki,ii], V10 ~ $[kLiIJ, Vloo ~ $1R[k~,in and there is iio E i=l i=l i=l {il, ... ,in} - {O} such that

N. Hirano and S. Rybicki

130

(a) iio:f:. gcdof {al, ... ,ad for any a!, . .. ,ak E {it,··· ,i;J,

(b) iio :f:. gcdof {aI, . .. , ad lor any aI, . .. , ak E {i;, ... , i;2 }, (c) m- ('V 2 f(O)PR[kio,iiol) :f:. ( -1

) m-('\72f(0)1V~»)+m-('\72f(00)1V2~»). m -

("'2f() v

. )

00 IIR[kio,JioJ

.

Proof. Suppose contrary to our claim that there are no nontrivial zeros of 'V I i.e. 'Vf-l(O) = {O}. Applying Splitting lemma at the origin to 'Vf we obtain number 0: > 0 and an Sl--equivariant gradient map 'Vfo: (cl(Do:(V)),So:(V)) -+ (V, 0) such that 1. 'VfO(Vl,V2) = ('V CPo (vt), 'V 2 f(0)(V2)) for (Vl,V2) E cl(Do:(V))n(VloE9V20),

2. 'Vcpo : (cl(DO:(VlO)),SO:(VlO)) -+ (VlO, VIO- {O}) is an Sl--equivariant gradient map. By the homotopy invariance of degree for Sl--equivariant gradient maps (see Theorem 3.9.(d) of [12]) and by Splitting lemma at the origin we obtain DEG('V f, Do: (V)) = DEG('V fo, Do:(V)). Applying Splitting lemma at infinity to 'V f we obtain numbers /31, /32 > 0 and Sl--equivariant gradient map 'V foo : (cl(D/lt (Vloo) x Dt32(V2oo)), 8(cl(Dt31 (Vloo) x Dt32(V2oo )))) -+ (V, V - {O}) such that 1. 'V foo (VI, V2)

=

('Vcpoo (vt), 'V 2 f( 00) (V2)) for (VI, V2) E cl(Dt31 (Vloo))

X

cl(Dt32 (V2oo)), 2. 'Vcpoo : (cl(Dt31 (Vloo)), St31 (Vloo)) -+ (Vloo, Vloo - {O}) is an Sl--equivariant gradient map. By the homotopy invariance of degree for Sl--equivariant gradient maps and by Splitting lemma at infinity we obtain

DEG('V f, Dt31 (Vloo) x Dt32 (V2oo ))

= DEG('V foo , Dt31 (Vloo) x Dt32(V2oo )).

Put n = Dt31 (Vloo) x Dt32 (V2oo ) - cl(Do:(V)). By Theorem 3.9.(c) of [12]) we obtain

Variational SI-symmetric problems with resonance at infinity

131

What is left is to show that D EG ("1, n) is a nontrivial element in U (SI ). More precisely, we will show that

is a nonzero integer. Applying Cartesian product formula we obtain

DEGz;.o ("1, Da(V)) = DEGz;.o ("1o, Da(V))

(3.2)

=

DEGz;.o (("CPo, ,,21(0)) , D a (VI0) x D a (V20))

=

DEG8 1("CPo, DaCViO)) . DEGz;.o (,,21(0), D a (V20))

+

DEG8 1(,,21(0), D a (V20)) . DEG z . ("CPo, D a (VI0)) "0

and

DEGz;.o ("1, D{31 (Vt») x D{32 (V2oo )) D EGz;.o ("100, D(31 (Vloo ) x D(32 (V2oo)) DEGz;;o (("CPoo, ,,21(00)),D{31(Vloo ) x D{32(V2oo ))

=

D EG 81 (" CPoo, D(31 (Vloo )) . D EGZ;.o (,,21 (00), D(32 (V2oo )) DEG 81 (,,21(00), D{32 (V2oo )) . DEGz;.o ("CPoo, D{31 (Vloo )).

+

(3.3)

By assumption 1. and by the definition of degree for SI-equivariant gradient maps we have

From assumption 2.(a) and 2.(b) it follows that the isotropy group of any point in V10 and in V100 is different from Zj.o" Therefore, by the definition of degree for SI-equivariant gradient maps we obtain DEGz, . ("CPo, D a (VI0)) = '0 oand DEG z . ("CPoo, D{31 (Vloo )) = O. Summing up, we obtain "0

and

N. Hirano and S. Rybicki

132

To compute DEGz jio (''V I, DO/(V)) and DEGz jio ('\11, D{31 (VlOO) X D{32 (V;OO)) it is enough to compute degree oflinear isomorphisms '\121(0)w20 and 2 I(oo)wr, respectively. Applying Corollary 4.3 of [12] we obtain

'\1

and

By (3.1), (3.4), (3.5) and assumption 2.(c) we obtain DEGz jio ('\11,0) :f O. From Theorem 3.9(a) of [12] it follows that '\11-1(0) n OZjio :f 0, a contradiction.

o The following two theorems are dual in some sense. Namely, assumptions 1. in these theorems are dual. But we should look at them a little bit more carefully. In Theorem 3.3 in assumption 2.(d) we consider case k = 2 because the Brouwer degree of a two-dimensional gradient map computed on a small disc centred at the origin is less or equal to 1. We do not consider case k = 2 in Theorem 3.4. The reason is that the Brouwer degree of a two-dimensional gradient map computed on a disc centred at the origin of a large radius can be equal to any integer. Theorem 3.3 Let '\11 E A.c7"(V). Assume that

n

~

i=l

i=l

~

2. V ~ EBIR[ki,ji], VlO ~ lR[k,O] E9 EBIR[k},jfl, Vloo ~ EBIR[k;,il] and i=l

there is jio E {il, ... , jn} - {O} such that

(a) jio :f gcdof {al'"'' ad for any al, .. ·, ak E {if, ... ,j;J, (b) jio :f gcdof {aI, ... , ak} for any at, ... ,ak E {j~, . . . , j~2}'

(c) if k = 1, then,· mfor, = 0,1,

('\1 1(0)IR[kio.jiol) 2

:f m-

('\1 ICOO)IR[kio.jiol) 2

Variational SI-symmetric problems with resonance at infinity (d)

133

il k = 2, then 2

2

(_I)m-(V /(O)wi'l)+m-(V /(oo)w200 ) .

'Y

'm- (V' 2 /(0)

...

IR[ k. o ,3'01

)..J.T

2

m- (V' /(00)IR[k'od'ol) lor'Y (e)

:s: 1,

il k > 2, then lor'Y

'Y . m- (V'2 I(O)IR[k,o ,j'ol)

= 0,1,2, . ...

:F m- (V'2 I( 00 )IR[k,o d'ol)

Proof. Suppose contrary to our claim that there are no nontrivial zeros of V'I i.e. V' 1-1 (0) = {O}. Repeating the first part of the proof of Theorem 3.2 we obtain positive numbers 0. , /31, /32 such that formulas (3.2), (3.3) hold true. Put fl = D{31 (Vl00 ) x D{32(V2OO ) - cl(Da(V)). By Theorem 3.9.(c) of [12]) we obtain

DEG(V'/, fl)

= DEG(V'/, D{31 (Vl

00

)

x D{32(V2OO ) ) ' - DEG(V'/, Da(V)). (3.6)

What is left is to show that DEG(V'/, fl) is a nontrivial element in U(SI). More precisely, we will show that

is a nonzero integer. By assumption 1. and by the definition of degree for SI-€quivariant gradient maps we have

DEGSI (V'cpo, D a (VI0)) E Z and DEGSI (V'cpoo, D{31 (~OO))

= 1.

From the assumption 2.(a) and 2.(b) it follows that the isotropy group of any point in Vlo and in Vl oo is different from Zj,o ' Therefore, by the definition of degree for 8 1 - equivariant gradient maps we obtain

Summing up,

and

N. Hirano and S. Rybicki

134

Notice that we have done reduction. In order to compute DEGz,.. (V I, D",(V)) '0 and DEGz"0 ·. (V I, DafJ 1(Vl00 ) x D/32(Vr)) it is enough to compute degree of a linear isomorphism V 2I (0) 1V20, V 2I (00) 1V2OO, respectively. Applying Corollary 4.3 of [12] we obtain

and (3.8) (3.9) If k = 1 then DEG s 1(Vt.po,D"'(V10)) = ±1,0. By (3.6), (3.7), (3.8) and assumption 2.(c) we obtain DEGz. (VI,n) f. 0. "0

If k = 2 then DEG S 1(Vt.po, D"'(VI0)) :s: 1. By (3.6), (3.7), (3.8) and assumption 2.(d) we obtain DEG z ·. (VI,n) f. 0. "0

If k > 2 then DEG S 1(Vt.po, D"'(VI0)) E Z. By (3.6), (3.7), (3.8) and assumption 2.(e) we obtain DEGz ·. (VI,n) f. 0. "0

From Theorem 3.9(a) of [12] it follows that V I-I (0) tion.

n nZjio f. 0, a contradic-

o Theorem 3.4 Let V I E ALi'" (V). Assume that

1.

TrO VI

s1

=

°

{}

n

• and dIm

OOS1 VI

= k..

~

~

2. V ~ EBIR[ki,ji]' y;O ~ EBIR[kI,jIJ, V100 ~ lR[k, 0] ill EBIR[k?,m and i=1

i=1

i=1

there is jio E {JI, ... ,in} - {a} such that (a) jio (b) iio

f. f.

gcdof {aI, ... • ak} for any aI, ... , ak E

{H,· .. ,j~1 },

gcdof {aI, ... ,ad for any aI, ... ,ak E

{i?, .. . ,j~2}'

(c) ifk = 1, thenm- (V21(0)I1R[kio,jioJ) for'Y = 0,1,

f.

'Y'm- (V21(00)IR[kiO,jiOJ)

(d) if k > 1. then m- (V2 I(O)IR[kiO,jiOJ) for'Y = 0, 1,2, ....

f.

'Y . m- (V2 I(OO)IR[kiO,jiOJ)

Variational SI-symmetric problems with resonance at infinity

135

Proof. Suppose contrary to our claim that there are no nontrivial zeros of j i.e. \1 j-l (0) = {O} . Repeating the first part of the proof of Theorem 3.2 we obtain positive numbers (x, /31, /32 such that formulas (3.2), (3.3) hold true. Put S1 = (D{31 (Vl00 ) x D{32(V200 )) -cl(D",(V)) . By Theorem 3.9.(c) of [12] we obtain

DEG(\1 j , S1) = DEG(\1 j, D{31 (Vl00 ) x D{32(V200 )) -DEG(\1 j, D",(V)). (3.10) What is left is to show that DEG(\1 j, S1) is a nontrivial element in U(SI) . More precisely, we will show that

DEGzj,o (\1 j, D{31 (Vl00 ) X D{32 (V2OO)) - DEGzj,o (\1 j, D",(V)) is a nonzero integer. By assumption 1. and by the definition of degree for SI--equivariant gradient maps we have

DEG 81 (\1'PO, D"'(VIO))

= 1 and DEG S 1(\1'Poo, D{31 (VlOO)) E Z.

From the assumption 2.(a) and 2.(b) it follows that the isotropy group of any point in Vl o and in Vl °O is different from Zi,o' Therefore, by the definition of degree for SI--equivariant gradient maps we obtain

DEGz"0 ·. (\1'Po,D",(ViO))

= 0 and DEG z 'to·. (\1'Poo,D{31 (Vl

00

))

= o.

Summing up, we obtain the following

DEGzj,o (\1 j, D",(V)) = DEGz;,o (\1 2j(O), D"'(V20)) and

DEG z 3'0 ·. (\1 j, D{31 (VlOO) X D{32 (V200 )) = D EG 81 (\1 'Poo, D (31 C~'lOO)) . D EGz;,o (\1 2j (00), D{32 (V200 )) . To compute DEGz;,o (\1 j, D",(V)) and DEGzj,o (\1 j, D{31 (Vl00 ) x D{32(V200 )) it is enough to compute degree of a linear isomorphism \1 2j(0)W20 and \1 2j(00)W2"'" respectively. Applying Corollary 4.3 of [12] we obtain 1

DEGzj,o (\1j,D",(V)) = 2 ·(-1)

2 m-(V !(OllY o ) 2

(2 ) ·m- \1 j(O)I1R[k'o,i'ol (3.11)

and

DEGz;,o (\1j,D{31(Vl OO ) x D{32(V200 ))

! . DEG S 1(\1'Poo, D{31 (Vl

00 )) •

=

(_l)m- (V2 !(oollY;") .

2

2 m- (\1 j(oo)IR[k'o,j'ol) .

(3.12)

N. Hirano and S. Rybicki

136

If k = 1 then DEG s ' (\1cpoo, Drh (VtO)) = ±1, O. By (3.10), (3.11), (3.12) and assumption 2.(c) we obtain DEGz,.. ('\l f, 0,) ::10. '0 If k > 1 then DEG s ' (\1cpoo, D{j, (Vl00 )) E Z. By (3.10), (3.11), (3.12) and assumption 2.(d) we obtain DEGZiiO (\1 f, 0,) ::I o. From Theorem 3.9(a) of [12J

it follows that \1 f- 1 (0) n O,Ziio ::I 0, a contradiction. Theorems 3.5 and 3.6 are similar. The only difference is that in Theorem 3.6

V10 s' can be nonzero linear space. Theorem 3.5 Let \1 f E AC7" (V) . Assume that 1. VIos'

= {O}

and V100 s'

= {O} ,

n

n,

i =1

i=1

2. V ~ EBlR[ki,j;j, V10 ~ EBlR[kLjIJ, l'too ~ lR[l,jioJ, 3. jio

::I gcdof {aI, . .. , ak}

for any aI, . . . ,ak E

{ii, · . . ,j~, },

4· 2

m- (\1 f(O)IIR[kio,jioJ)

(-1) for'Y

::I

(2

m-(V2f(0)JVo)+m-(V2f(oo)JVOO) 2

2

•

m- \1 f(oo)IR[kio.JioJ

) + 'Y,

= ±2, O.

Then, \1f-l(O) n (VZiio - {O}) ::10. Proof. Suppose contrary to our claim that there are no nontrivial zeros of \1 f i.e. \1 f- 1 (0) = {O} . Repeating the first part of the proof of Theorem 3.2 we obtain positive numbers cr., (31 ,/32 such that formulas (3.2), (3.3) hold true. Put 0, = D{j, (Vl00 ) x D{j2(V2OO) - cl(Da(V)). By Theorem 3.9.(c) of [12]) we obtain

What is left is to show that DEG(\1 f, 0,) is a nontrivial element in U(8 1 ) . More precisely, we will show that

is a nonzero integer. By assumption 1. and by the definition of degree for 8 1 -equivariant gradient maps we have

Variational Sl-symmetric problems with resonance at infinity

137

From assumption 3. it follows that the isotropy group of any point in V10 is different from Zjio' Therefore, by the definition of degree for Sl---€quivariant gradient maps we obtain DEGz jio (vripo, Do< (V10)) = O. Summing up,

We have done reduction. In order to compute D EGz , (vr I, Do< (V)) it is "0 enough to compute degree of a linear isomorphism vr21(O)W20 . In order to compute DEGz"0 " (vrl,D{31(V100 ) x D{32(V2OO)) it is enough to compute degree of a linear isomorphism vr21(oo)W2oo and degree of vripoo' Applying Corollary 4.3 of [12] we obtain

DEGz " (vrl, D{31 (V100 ) x D{32(V2OO )) "0

=

(_I)m- (V2 /(oo)wr) . ~ . m- (vr 2 I(OO)IR[kio,jiol)

+

(3.15)

oo 00 ( _I)m- (V2 /(oo)w2 ) . DEGz""0 (vripoo, D{31 (V1 )). From assumption 2. ana Lemma 2.16 of [13] it follows that 00 DEG zjio (vripoo, D{31 (V1 )) = ±1, O. By (3.13), (3.14), (3.15) and assumption 4. we obtain DEGz" , (vr I, n) '" O. From Theorem 3.9(a) of [12] it follows that '0

1- 1 (0) n nZjio

'"

0, a contradiction.

o

Theorem 3.6 Let vr 1 E AC~ (V). Assume that

n

nl

2. V:::::l ffilR[ki,ii], V10 :::::l1R[k,O] ffi ffilR[kl,ifl, V100 :::::l1R[I,iio], i=l

i=l

N. Hirano and S. Rybicki

138

3. jio f:. gcdof {a1, . .. , ak} for any a1 , . .. , ak E {jt,· .. ,j~l },

4.

/'1 · m- ('\72/(0)I1R[k;0,j;0)) f:. m- ('\72/(00)I1R[k;0,j,0)) +/'2, where

(a) if k

= 1,

(b) if k

> 2, then /'1 E Z and /'2

then /'1

= ±1 , 0

and /'2

= ±2,0,

= ±2,

°

or /'1

. (_1)m-(V'2/(O)IV~»)+m-(v2/(00)1V200). m -

m-

('\7 2 I( 00 )IIR[k;o ,j;o))

where k

= 2, /'1

~

('\7 2/(0)

...

IIR[k'o,bo)

)...J. -r

+ /'2,

1 and /'2

= ±2,0.

Proof. Suppose contrary to our claim that there is no nontrivial zeros of '\71 i.e. '\71- 1 (0) = {a} . Repeating the first part of the proof of Theorem 3.2 we obtain positive numbers a., /31, /32 such that formulas (3.2), (3.3) hold true. Put fl = (D/3, (Vl00 ) x D/32 (V2OO)) -cl(D",(V)) . By Theorem 3.9.(c) of [12]) we obtain

DEG('\7 I, fl)

= DEG('\7 I, D/3, (V1OO) x D/32 (V2OO)) -DEG('\7 I, D",(V)).

(3.16)

What is left is to show that DEG('\7 I, fl) is a nontrivial element in U(SI). More precisely, we will show that

is a nonzero integer. By assumption 1. and by the definition of degree for S1--equivariant gradient maps we have

By assumption 3. the isotropy group of any point in V10 is different from Zj,o· Therefore, by definition of degree for SI--equivariant gradient maps we obtain DEGz,·. ('\77 ~

1(5')-' Hb+p(n) (En)

Marek Izydorek

156

2.5. Definition. The q-th Borel cohomology group of a G-spectrum E E(~)

=

is the inverse limit group

H~(E) := lim{Hb+p(n) (En), hh+p(n)} . r

2.6. Remark. Directly from the definition one has Ha(E) ~ Hb+p(n) (En), for n large enough. It is also easily seen, that Ha(E) := ffiqEZH~(E) admits H*(BG)- module structure. A G-map of spectra f : E -+ F induces a homomorphism fa : Ha(F) -+ Ha(E) of H*(BG)-modules . Let f : H -+ H be G-£S-vector field . Let X be an isolating G-neighbourhood for the local flow 1} generated by f, and let Ex stands for corresponding Gspectrum. 2.7. Definition. We say that S

= Inv(X,1})

is homologically pt-hyperbolic

of index p E Z if there exists u E H~ ( Ex) , so that the map H*(BG)

-+ Ha(Ex) : a -+ a * u

is an isomorphism. The notion of homologically hyperbolic (*-hyperbolic) isolated G- and 1}invariant sets in a locally compact spaces has been introduced by Floer [11] in more general context. However, the notion of homological pt-hyperbolicity is enough for our considerations.

3

Hamiltonian systems

Let us first recall a general setting which will be used in our considerations. Given a Hamiltonian function Q E C 1 (JR 2m,JR) consider the Hamiltonian system of differential equations

i = J\lQ(z)

where J

=

(3.1)

[0 -I]

l O i s the standard symplectic matrix and \1 denotes the

gradient with respect to z E JR2m . We will study multiplicity results for 271'periodic solutions of (3.1) . Let us denote by H = Ht(Sl , JR2m) the Hilbert

Multiplicity results for Hamiltonian systems

157

space of 2rr-periodic, 1R2m -valued functions 00

z(t)

= ao + L (an cos(nt) + bn sin(nt»,

where ao, an, bn E 1R2m

n=1

with the inner product given by (3.2) where < a, b > denotes the standard inner product in 1R2m . Here and subsequently S1 is identified with the quotient group IRj2rrZ. Define an action of S1 on H by the time shif, Le.: (gz)(t) := z(t + g) z E H, t, 9 E S1 . Clearly, that action is linear and invariant with respect to the inner product (3.2). Hence, H is an orthogonal representation of S1 . If II "\1Q(z) II~ C1 + C2· II Z 11 8 for every z E 1R2m and some positive 5, then z(t) is a 2rr-periodic solution of (3.1) if and only if it is a critical point of the functional 4> E C 1 (H, 1R) defined by

4>(z) =

1

-2 < Lz ,z >H -¢(z),

where

< Lz,z >H=

121< < Ji(t),z(t) > dt

and

¢(z) =

121< Q(z(t»dt.

(3 .3)

(cf. [17)). It follows from the above that 4> is S1-invariant, i.e.: 4>(gz) = 4>(z) for every 9 E S1 and z E H. Hence, "\14> is an S1-map. Consequently, if z E H is a critical point of 4> then the whole orbit S1 z = {gz E H ; 9 E S1} consists of critical points. It is shown in [17] that the mapping "\1 ¢ is compact and therefore -"\14> : H --+ H is a vector field which can be written in the form -"\14>(z)

= Lz + K(z)

where K : H --+ H is completely continuous. However, K = -"\1 ¢ may not be loco Lipschitz continuous. Fortunately, it can be then replaced by a completely continuous and loco Lipschitz continuous S1-map K : H --+ H so that, J : H --+ H, J(z) = Lz + K(z) is a pseudo-gradient vector field for the functional -4> (see [6]) . Thus, with no loss we can assume, that "\14> is loc o Lipschitz continuous.

Marek Izydorek

158

Choose e1, .. . , eZ m the standard basis in IR 2m and denote Ho

= span {e1, ...

, e2m}

H:;

= span{(cos(nt))ej + (sin(nt))Jej : j = 1, ...

H;;

= span{(cos(nt))ej -

(sin(nt))Jej : j

. 2m},

n E N,

= 1, .. . ,2m},

n E N.

Clearly, H:; and H;; are S1-representations for every n E N. It is seen from (3.2), (3.3) that L is a differential operator in H which is explicitly given by 00

Lz =

L Jbn cos(nt) -

Jan sin(nt).

n=1 So, Lz

= 0 if z is a constant function Lz

={

-z

,

and

z E H:;

z , z

E

H;;

n

= 1,2, .. .

Put Hn = H:; ffi H;;, n = 1,2, ... Obviously, H = ffi~=oHn, spaces Hn are mutually orthogonal representations of S1 and Ho = ker L. For more details in non-equivariant case we refer the reader to [19] . Thus we conclude, that - V' is an S1-'cS-vector field. Moreover. ~ = (H;;+d~=o is a sequence of S1-representations which is needed to define S1-

spectra corresponding to isolating S1-neighbourhoods. Finally, if A is a symmetric 2m x 2m-matrix and Q(z) (3.1) becomes a linear Hamiltonian system

= ~ < Az, z >, then

= JAz

i

The vector field V' : H -+ H corresponding to that system preserves all spaces Hn and the restriction of V' to Hn , n ;::: 1 . may be identified with the linear map on IR4m whose matrix is Tn (A)

=[

-.!.A nJ

-J]

-~A

(3.4)

and with -A on IR2 m if n = 0 (see [19]). The following numbers have been defined by Amann and Zehnder (see [1]): 00

i-(A) := M-(-A)

+ L(M-(Tn(A)) n=l

- 2m)

Multiplicity results for Hamiltonian systems

159

00

iO(A) := MO( -A)

+

L MO(Tn(A)) n=1

where M- (U) is the number (with multiplicity) of negative eigenvalues of a symmetric matrix U and MO(U) is the dimension of its kernel. In the sequel we will use SI-equivariant Borel cohomologies with real coefficientes. We recall, that H*(BS1,IR)::::::J IR[w], w E H2(BS1,IR). Denote by ", a local flow generated by - V 4? 3.1. Proposition. Assume that Q(z)

> 0,

D(r) := {z E Hj

with Inv(D(r),,,,) PROOF:

is linear

II Vq(z) 11= o(lzl)) as z -+ O. If iO(A) = 0 then for sufficiently II z II~ r} is an isolating SI-neighbourhood for",

symmetric and small r

= ! < Az, z > +q(z), where A

= {O} and {O} is homologically pt-hyperbolic of index i-(A).

It follows from our assumptions, that an SI-CS-vector field - V4? :

H -+ H has a derivative Ao at 0 E H defined by (3.4) which is an isomorphism since iO(A) = O. Thus, the origin in H is an isolated invariant set and D(r) is an isolating SI-neighbourhood for TJ with Inv(D(r), TJ) = {O} if r is sufficiently small and positive. Moreover, the SI-CS-index h~~(D(r),TJ) is the homotopy type of a spectrum E = E(~) such that (i) for each n ~ 0, En is a one-point-compactification of a certain finite-dimensional representation of SI and therefore it is a spherej (ii) En+l = SH;:+l "En and dim En = i-(A) +2mn > 0 for sufficiently large n, say n

~

r.

Consequently, by Remark 2.6

HSI (E, IR) Since dim Er = i- (A) modules

::::::J

H;tP(r)(Er, IR) = H;t 2rm (Er , IR).

+ 2mr

there is the suspension isomorphism of IR[w]-

where So stands for zero-dimensional sphere with a base point. Hence,

which proves that {O} is homologically pt-hyperbolic of index i-(A).

Marek Izydorek

160 3.2. Proposition. Assume that Q(z)

= ~ < Az, z > +q(z)

where A is linear

symmetric and \7q(z) is bounded. If iD(A) = 0 then there exists a maximal bounded isolated invariant set T for.". It is homologically pt-hyperbolic of index i-(B).

The proof is similar to that of Proposition 3.1. The following theorem has been proved in [15] . 3.3. Theorem. Let." be a flow generated by a gradient 5 1-CS-vector field f : H ~ H, i. e.: f = \7 h for some 5 1-invariant functional h : H ~ lit Assume that:

(1) {O} cHis homologically pt-hyperbolic isolated invariant set for

'T/ of

index 2p E Zj (2) there is a homologically pt-hyperbolic isolated (G- and .,,-) invariant set T for." of index 2q E Z and 0 E Tj (3) if x E Tn f- 1(0) and x =J. 0 then its isotropy group G", =J. 51. Then, except 0 there are at least I p - q I orbits in Tn f-1(0).

As a direct consequence of the above Theorem and Propositions 3.1 and 3.2 we obtain the following. 3.4. Theorem. Let A, B be symmetric, 2m x 2m-matrices. Let Q E C1 (JR 2m , JR) satisfy the following conditions: • Q(z) = ~ \7qD(Z) 11= • Q(z) = ~ bounded;


+qD(Z) in a neighbourhood of 0 E as z ~ OJ

and

II

< Bz, z > +qoo(z) in a neighbourhood of 00, and \7qoo is

• 0 E ]R2m is the unique critical point of Q.

If iD(A)

]R2m,

= iD(B) = 0, then system (3.1) possesses at least 1

2 I i-CAl nonconstant 27r-periodic solutions.

i-(B)

I

Multiplicity results for Hamiltonian systems

161

The above theorem generalizes Costa and Willem result from [7] obtained for positive definite matices A, B and for strictly convex Hamiltonian functions (see also [16]). In particular, 0 E JR2m is always a global minimum of Q. In our case it may be a saddle point. In what follows U stands for an open halfline (0,00), where 0 is negative or -00. Consider a map F : U -+ JR of class C 1 such that

(F.l) F'(x)

i

0 for each x E U ;

(F.2) there exists lim",-+oo F'(x) (finite or infinite); (F.3) there exists F"(O). Let B : JR2m -+ JR2m be a linear isomorphism. We will be concerned with Hamiltonian functions Q : JR2m -+ JR,

Q(z) Since the case lim",-+oo F'(x)

=a i

= F(II B

11 2 ) .

(3.5)

0 is rather standard we discuss the most

interesting extreme cases only, i.e. lim F'(x) = 0 and lim F'(x) = x-+oo

x~oo

±oo.

Our first theorem states as follows.

3.5. Theorem. Let a Hamiltonian function Q be of the form {3.5} with F satisfying lim",-+oo F'(x)

= O.

Then, system {3.1} possesses at least

. ! I i-(2F'(0)BT B) - 2m I if F'(O) > 0, . ! I i-(2F'(0)BT B) I if F'(O) < 0, nonconstant 27r-periodic solutions. PROOF:

By our assumptions Q is continuously differentiable in JR2m and

VQ(z)

= 2F'(11 Bz 112)BT Bz,

where BT denotes the map conjugated to B. Since for each x E U F'(x) one has

VQ(z)

= 0 iff z =

O.

i 0,

Marek lzydorek

162

Furthermore, the vector field \1Q : JR2m -+ JR2m is assymptotically linear, i.e. admits derivatives at the origin and at the infinity. The derivative at 0 E JR2m

is selfadjoint and positively or negatively definite , depending on a sign of

F'(O). We claim that the derivative at the infinity Aoo = 0 so that, we have a resonance. It is enough to show that lim z-+oo

II \1Q(z) II = o. II z II

Indeed, we have

0< lim 112F'(11 Bz 112)BT Bz II < 2 II BT B II lim IF'(II Bz 112)1 II z II

II z II

- z-+oo

-

II z II

Z-+oo

= o.

Given f3 E JR such that F'(x) = f3 for some x> O. Set y = max{x E U; F'(x) f3} and define F{3 : U -+ JR, Ii: x _ {

{3()-

F(x) F(y)+f3(x-y)

=

if x:::; y if x> y

Then, F{3 is a CI-map and if x:::; y if x> y The gradient of the modyfied Hamiltonian function Q{3(z) given by the formula

\1Q (z)

{3

={

\1Q(z) 2f3BT B

if if

= F{3(11

Bz

II Bz 112:::; Y II Bz 112> y

W)

is

(3.6)

Obviously, \1Q{3(z) is assymptotically linear and its derivatives at 0 E JR2m and at the infinity are equal to Ag respectively. Since limx--+oo F' (x) choose f3 so close to 0 that

iO(A~) Moreover, as f3F'(O)

= Ao = 2F'(O)BT B

= 0 and BT B

and A~

= 2f3BTB,

is positively definite one can

= 0 and i-(A~,) = M-(-2f3B T B).

> 0 one has M-(-2f3B T B)

= M-(-2F'(O)BTB).

Multiplicity results for Hamiltonian systems

163

If iO(Ao) = 0 then by Theorem 3.4 the Hamiltonian system i

= J\1Q/3( z )

(3.7)

possesses at least

nonconstant 2rr-periodic solutions. As iO(A~) = 0 the system

does not have nonconstant 2rr-periodic solutions and therefore each 2rr-periodic solution of (3.7) is a solution of (3.1). Finally, let us notice that M-( -2F'(0)B T B) is equal to 0 if F'(O) < 0 and is equal to 2m if F'(O) > O. There are many examples among classical functions satisfying assumptions of Theorem 3.5. As a map F(x) one can choose for instance: ce-"', cln(1 + x), carctg(x) , c(1

+ x)-S

with s > 0 or - 1 < s < 0, ect.,

where c -:F O. Note, that if F is one of the above F(x 2 ) is non-convex so that,

Q(z)

= F(II Bz 112) in non-convex as well. Choose F(x) = ~e-'" and B = aId, where Id is the identity map and k < a 2 < k + 1 for some kEN U {O} then

EXAMPLE :

in

1R2m

Q(z) = !e-lIazIl2 2

and system (3.1) possesses at least k . 2m nonconstant 2rr-periodic solutions.

3.6. Theorem. Let a Hamiltonian function Q be of the form (3.5) with F satisfying lim"'-too F'(x) = 00. Then, system (3.1) possesses infinitely many nonconstant 2rr-periodic solutions. We use similar methods as in the proof of Theorem 3.5. This time however, \1Q is not asymptotically linear. By our assumptions F'(x) > 0 for every x E U. Since the Hessian of Q at 0 E 1R2m , A o = 2F' (O)BT B is positively definite it follows from (3.4) that there is a increasing sequence of real numbers PROOF:

«(3n)'f such that: • limn-too (3n =

00 ;

Marek Izydorek

164 • iO(f3nBT B)

= 0 for every n

• lim n -+ oo i- (f3nBT B)

E N;

= 00.

By Theorem 3.4, if iO(A o) = 0 then for each n E N the Hamiltonian system (3.8)

i = J"lQ(3n (z)

possesses at least

!2 I i-(Ao) -

i-(f3n BTB ) I

nonconstant 21T-periodic solutions, where Q(3 is defined by (3.6) . The condition iO(f3nBT B) = 0 implies that all 21T-periodic solutions of (3.8) are also solutions of (3.1). Since lim n -+ oo li-(f3nBTB)1 = 00 our assertion is concluded. The case lim x -+ oo f'(x) = -00 is similar. As an example one can choose f(x) = (1 + X)8 with s > 1, f(x) = eX ect. Clearly, there are also functions for which the corresponding Hamiltonian

x3

function is non-convex, e.g.: f(x) = 2x - 3!

x5

+ ST·

References [1) H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscr. Math. 32, (1980), 149-189. [2) T . Bartsch, "Topological methods for variational problems with symmetries"; Lecture Notes in Math. 1560 Springer-Verlag, Berlin, Heidelberg, 1993. [3) T . Bartsch and M. Clapp, Critical point theory for indefinite functionals with symmetries, J. F'unct. Anal. 138 No .1, (1996), 107-136. [4] T . Bartsch and M. Willem, Periodic solutions of non-autonomous Hamiltonian systems with symmetries, J. Reine und Angew. Math. [5) V. Bend, On critical point theory for indefinite functionals in the presence of symmetries, Trans. AMS 274, (1982), 533-572.

[6) K.C . Chang, "Infinite Dimensional Morse Theory and multiple Solution Problem", Birk- hauser, Boston, (1993) . [7) D.G. Costa and M. Willem, Lusternik-Schnirelman theory and asymptotically linear Hamiltonian systems, Colloquia Math. Soc. J . Bolyai vol. 47 Differential Equations: Qualitative theory, Szeged Hungary 1984, North Holland 1986, 179191.

Multiplicity results for Hamiltonian systems

165

[8] T. tom Dieck, "Transformation groups", W . de Gruyter and Co., Berlin, 1987. [9] E. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978) 139-174. [10] G. Fei, Relative Morse Index and Its Application to Hamiltonian Systems in the Presence of Symmetries, J . Diff. Equat., 122, (1995), 302-315. [11] A.Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets; Ergod. Th. & Dynam. Sys. 7 (1987), 93-103. [12] A. Floer and E. Zehnder, The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems, Ergod. Th. & Dynam. Sys. 8' (1988), 87-97. [13] K. G-:ba, M. Izydorek and A. Pruszko, The Conley Index in Hilbert Spaces, Studia Math. 134 (3), (1999), 217-233. [14] M.lzydorek, A cohomological Conley index in Hilbert spaces and applications to strongly infinite problems, accepted in J . Diff. Equat. [15] M.lzydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems, submitted. [16] J. Mawhin and M. Willem, "Critical point theory and Hamiltonian systems" , Berlin Heidelberg New York, Springer-Verlag, (1989) . [17] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. Conf. Ser. Math., vol 35, Providence, R.I., Am. Math. Soc. (1986). [18] E. H. Spanier, "Algebraic topology", New York, Mc Graw-Hill 1966. [19] A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209, (1992), 375-418. [20] Z.Q.Wang, Equivariant Morse theory for isolated critical orbits and its applications to nonlinear problems; Lect. Notes in Math. 1306, Springer-Verlag (1988),

202-221. Technical University of Gdansk, Faculty of Technical Physics and Applied Mathematics, 80-952 Gdansk, ul. G. Narutowicza 11/12, Poland current address: Universitat Rostock, Fachbereich Mathematik, Universitatsplatz 1, 18055 Rostock, Germany email: [email protected]

The Dynamics of the Flow for Prescribed Harmonic Mean Curvature * Huai-Yu Jian and Bin-Heng Song Department of Mathematical Sciences Tsinghua University, Beijing 100084, P.R.China (e-mail: [email protected])

Abstract In this article, we study the dynamical behavior of a parabolic equation on supplement a proof of the geometric lemma in [8] and give a new and more direct proof of the apriori estimate which is essential for the existence results of convex hypersurfaces with prescribed harmonic mean curvature in [8].

sn,

1. Introduction Let M be a smooth embedded hypersurface in R n+1 and k1 , k2 , •.. ,kn be its principal curvatures. Then H- 1 is called the harmonic mean curvature of M if 1 1 1 (1.1) , H= -+ - + ... +-. kl k2 kn The question which we are concerned with is that given a function in Rn+1, under what conditions does the equation

J defined

H-1(X) = J(X),X E M has a solution for a smooth, closed, convex and embedded hypersurface M, where X is a position vector on M. The kind of such questions was proposed by S.T.Yau in his famous problem section (1). Many authors have studied the cases of mean curvature and Gauss curvature instead. See, for instance, [2, 3) for the mean curvature and [4, 5) for the Gauss curvature, and [6, 7) for general curvature functions . ·Supported by National Natural Science F'undation of China (Grant No. 19701018)

166

Flow for prescribed harmonic mean curvature

167

Let F = 1-1, then the problem above is equivalent to looking for a smooth, closed, convex and embedded hypersurface M in Rn+l such that

H(X) = F(X) , X E M

(1.2)

where H is the inverse of harmonic mean curvature given by Eq. (1.1). We are interested only in the hypersurfaces MeA, a ring domain defined by

for some constants R2 > Rl > O. For this purpose, we need to suppose that F is a smooth positive function defined in Rn+l satisfying (a) F(X) ~ nR2 for IXI = R2 and F(X) < nR1 for IXI = R 1 , and (b) F is concave in A. In [8], the first author used a heat flow method to deform convex hypersufaces to a solution to Eq. (1.2). This leads us to consider the following parabolic equation

~~ = (H(X) - F(X)) v(X) , X E Mt , t E (0, T) } X(·,O) given,

(1.3)

where X(x, t) : sn --+ Rn+l is the parametrization of M t given by inverse Gauss map, which will be solved, and v(X) is the outer normal at X E M t , so v(X(x, t)) = x by the definition. We use Mo to denote the given initial hypersurface. The following is one of the main results in paper [8]. Theorem 1.1. Suppose that F is a smooth positive function satisfying conditions (a) and (b), and the initial hypersurface Mo C A is smooth, uniformly convex and embedded, satisfyingH(Xo) ~ F(Xo) for all Xo E Mo . Then equation (1.3) has a unique smooth solution for T = 00 which parametrizes a family of smooth,closed,uniformly convex and embedded hpersufaces, {Mt : t E [O,oo)}. Moreover, there exists a subsequence tk --+ 00 such that Mtk converges to a smooth,closed,uniformly convex and embedded hypersurface which lies in A and solves problem (1 .2). There are two key facts in the proof of Theorem 1.1. One is Lemma 2.3 in [8] . We will call it Geometric Lemma. Since its proof was omitted in [8] due to the limited pages, We are going to supplement it in the next section. The other is the apriori estimate for the solution, IX(x, t)12 , of Eq. (1.3). Also see section 2 in [8] . We will give a new and concise proof for this estimate.

Huai- Yu Jian and Bin-Heng Song

168

2. A Geometric Lemma In this section, we will supplement the proof of Lemma 2.3 in [8]. Lemma 2.1 Let X be the positive vector of a smooth,closed hypersurface M in Rn+l with outer normal veX) at X E M . Then if IXI =< X, X > ~ attains a maximun R at a point Xo E M, then Xo = Rv(Xo) and

II(w, w)

~

1

/ig(w, w), Vw E TXoMj

if IXI attains a minimum r at a point Xo E M, then Xo

II(w,w)

~

= rv(Xo) and

1

-g(w,w), Vw E TXoM, r

where 9 is the metric on M and II is the second fundamental form of M with respect to the direction -v. Proof. We consider only the first case, because the latter is completely anagolous. For Xo E M and any w E TXoM, choose a curve ')'(s) on M,,,( : [0,1] -+ M, such that "(0) = X o, l' = w. Let p(X)

= IXI. Since p2(X) attains a maximum at Xo \lp2 = 0 and \l2p2

~

E M, then at this point

O.

Therefore, we have

and

~

ds 2P2 (')'(O)) = 1'(0)· \l2p2(XO)' 1'(0) + \lp2(Xo)' ;yeO) ~ O. On the other hand,

and

1 d2p2

.

2

..

2 ds2 = I')'(s) I + ')'(s) . ')'(s). Thus,

o = 2')'(0) . 1'(0) = 2Xo . w

(2.1)

and

o ~ 11'(0)12 + Xo

. ;yeO).

(2.2)

Flow for prescribed harmonic mean curvature

169

Since w E TXoM can be arbitrary, Eq. (2.1) implies

Xo

= IXolv(Xo) = Rv(Xo),

therefore

Xo . 1'(0)

=

R < v(Xo),1'(O) >

=

R < v(Xo), Dt(o)1'(O) > -R II ("y(0), 1'(0)) -R II (w, w),

which, together with Eq. (2.2), gives us that

II(w,w)

>

~h(OW R 1 R

< 1'(0),1'(0) >

1 R

< Dt(o)X, Dt(o)X >

1 "Rg("y(O),1'(O)) 1 "Rg(w,w).

3. A New Proof of the Apriori Estimate In this section, we will provide a new and concise proof of the apriori estimate for Eq.(1.3), which is essentially important for the proof of Theorem 1.1. We recall some facts in (9). Let el, e2, ... , en be a smooth local orthonormal frame on sn, and let 'Vi = 'Vei) i=l, 2, ... , n and 'V = ('VI, 'V 2,· .. , 'V n ) be the covariant derivatives and the gradient operator on sn, respctively. Since X(x, t) is the inverse Gauss map, the support function of M t is given by

>, x

E sn

(3.1)

where < ., . > denotes the usual inner product in tal form of M t is

Rn+l.

The second fundamen-

U(x, t) =< x, X(x, t)

hij(X (x,

t)) =< 'V iX, 'V jX >= 'Vi 'V jU(X, t) +c5ij u(x, t), i, j = 1,2, ... , n. (3.2)

IT M t is strictly convex, then h ij is invertible, and hence the inverse harmonic mean curvature is the sum of all the eigenvalues of the matrix bij = [ g''k hkj )-1

,

Huai- Yu Jian and Bin-Heng Song

170

where gij is the metric of M t . But the Gauss-Weingarten relation

'ViX = hikl/'V/x and the fact < 'Vix, 'Vjx

>= Oij imply gij = hikhjk.

1 H = -

1

+ ... + -

k1

kn

Therefore, bij

= ~u + nu,

= hij and (3.3)

where ~ = 2::7=1 'Vi'V i · We will often use the fact that x, 'V 1 x, 'Vzx ,··· , and 'V nX form a standard orthonormal basis at point X (x, t). It implies

X(x , t)

= u(x, t)x + 'Viu(x, t)'Vix

(3.4)

and

IX(x, tW

= UZ(x, t) + l'Vu(x, tW o

(3.5)

From now on, we assume that the initial hypersurface Mo is smooth, closed, strictly convex and satisfies H(Xo) 2: F(Xo) for all Xo E Mo. That is Uo E coo(sn) and for some positive constant Co, one has (3.6)

and ~uo(x)

+ nuo(x) - F(uo + 'ViUO'ViX) 2: 0, "Ix

E

sn,

(3.7)

Where I denotes the n x n unit matrix and Uo is the support function of Mo . The following lemma is well known (see [8]). Lemma 3.1. If for t E [0, T) with T ~ 00, X(x, t) is a solution of (1.3) which parametrizes a smooth, closed, strictly convex and embedded hypersurface, then the support functions u(x, t) of M t satisfy

~~ = ~u + nu - F(ux u(·,O) = uo(-)

+ 'ViU'ViX) , (x, t) E sn

x (0, T) }

(3.8)

and

'VZu + uI > 0 in

sn

x (0, T) .

(3.9)

Conversely, if u is a smooth solution to (3.8) and satisfies (3.9), then the bypersurface M t , determined by its support function u(x, t), is a smootb, closed, strictly convex, embedded hypersurface and solves (1.3) for t E [0, T).

Next, we sup})ose that u is the solution to Eq. (3.8) on time interval (0, T) with the initial data satisfying Eqs. (3.6), (3.7) and

Mo C A and u(x.O) = uo(x) > Rl "Ix E

sn

(3.10)

171

Flow for prescribed harmonic mean curvature We will estimate u and its derivatives. Lemma 3.2. For all (x, t) E sn x (0, T), we have

°

au(x, t) at ~.

(3.11)

Proof. Let

= ~u(x, t) + nu(x, t) -

G(x, t)

F(u(x, t)x

+ V'iUV'iX).

(3 .12)

Using Eqs.(3.4) and (3.8), we compute that for t E (0, T), aG at

= =

au

au

au ·

au

·

+ n at - Fj(X)( at Xl + V'i at V'i Xl ) ~G + nG - Fj(X)(Gxj + V'iGV'ixj), ~ at

(3.13)

°

a%f;).

where Fj = But condition (3.7) says that G(x, 0) ~ for all x E sn. This means that G(x, t) is a supersolution to the equation of type (3.13) with zero initial data. By comparison principle [10], we have G(x, t) ~ for all (x, t) E sn x (0, T). Using equation (3.8) again, we have obtained (3.11) .

°

Theorem 3.3. Under assumptions (3.6), (3.7) and (3.10), we have R~ ~ u 2(x , t)

+ lV'u(x, t)12 < R~, 'V(x, t)

E

sn

x (0, T).

(3.14)

Proof. With the aid of (3.11) and (3.10), we see that u(x, t) ~ Rl

> 0, 'V(x, t)

E

sn x [0, T).

(3.15)

This immediately implies the left hand side inequality of (3.14). Next, we will prove its right side one, i.e., u 2(x, t)

+ lV'u(x, tW < R~, 'V(x, t)

E

sn

x (0, T).

(3 .16)

For this purpose, we compute, using (3.4) and (3.8), that

+ 2nu 2 - 2F(X)u - 21V'u1 2 + 2nu 2 - 2F(X)u,

2~uu

= and

alV'ul at

2

=

~U2

(3.17)

Huai- Yu Jian and Bin-Heng Song

172

Observing that AI\7uI 2

= =

2\7i(\7i\7kU · \7kU) 21\7\7uI 2 + 2\7iUA \7i u

and using the standard formula for interchanging the order of covariant differ(see [ll]) entiation with respect to the orthonormal frame on

sn

\7 i Au

= A\7 i u -

(n - l)\7iu,

(3.19)

we obtain from (3.18) that

81~:12 = AI\7uI 2 _ 21\7\7uI 2 + 21\7u1 2 -

2\7 i uFj(X)\7 i xj .

(3 .20)

On the other hand, by (3.4), (3.2) and the fact that x, \71X,·· ·, \7 n x form a standard basis, we have < \7 i X, x >= 0 and

= =

\7 i X

< \7 i X , \7jX > \7jX h ij \7 j x

Thus, we have 2\7i UFj(X)\7i Xj

= =

Fj(X)(2\7 i u\7;\7 kU + 2u\7 kU)\7 kxj Fj(X)\7k(l\7uI 2 +U2)\7kxj.

(3.21)

Putting (3.21) in (3.20) yields 81\7u1 2

---at =

2 AI\7uI - 21\7\7uI 2 + 21\7u1 2 - Fj(X)\7 k (U 2 + l\7uI 2)\7 kxj,

which, combined with (3.17), implies

where we have used (3.5) . Now , let

f(t)

= xESn max IX(x, t)12,

t

E

[0, T) .

Were inequality (3.16) not true, then by the assumption Mo C A (see (3.10)) we could find t2 E [0, T) such that f(h) = ~ . Let

tl = inf{t E (O,T) : f(t) =

RD

Flow for prescribed harmonic mean curvature

173

and choose Xl E sn such that J(tt}

= IX(XI' tl}1 2 = R~ .

(3.23)

Obviouly, IX(XI,t}1 2 < IX(XI,tIW for all t E [O,tt}, so that we have 81XI 2 at(XI,tt} ~ O.

(3.24)

On the other hand, since (Xl, tt) is the maximum point for the function IX (x, tl W with respect to (x E sn), we have 61XI ~ 0 and V'IXI 2 2

=0

at (Xl, tt)'

Therefore, Eq. (3.22) yields 2

81XI a t

~

2u(nu - F(X}} at (XI,tt).

(3.25)

Furthermore, it follows from (3.23) and condition (a) that

F(X(XI' tt}} > nlX(xI' tt}l ~ nlu(xl, tt}l .

(3.26)

With the aid of (3.15) and (3.25), we see that (3.26) turns to 81XI 2 at(XI,tl } < 0,

which yields a contradiction with (3.24) . In this way, we have proved inequality (3.16). Acknowledgement. The first author would like to thank professors R . Bartnik, K. Tso, W. Y . Ding, and Y.D . Wang for many helpful conservations and professor B . Chow for his interests in this work.

References 1. S. T . Yau, in Seminar on differential geometry, ed. S. T . Yau ( Ann of Math Stud 102, Princeton, 1982), pp. 669-706. 2. A. Treibergs and S. W. Wei, J. Differ. Geom. 18 (1983), 513-521. 3. K. Tso, Ann. Scuola Norm. Sup. Pisa 16 (1989), 225-243. 4. V. 1. Oliver, Trans. Amer. Math. Soc. 295 (1986), 291-303. 5. K. Tso, J. Differ. Geom. 32 (1991), 389-410.

174

Huai- Yu Jian and Bin-Heng Song

6. 1. Caffarelli, J . Nirenberg and J. Spruck, in Current topics in partial differential equations, eds. K. Ohya, K. Kasahara and N. Shimakura

7. 8. 9. 10. 11.

(Kinokunize Co., Tokyo, 1986), pp. 1-26. C. Gerhardt, J. Differ. Geom. 43 (1996), 612-641. H. Jian, Science in China Ser. A 42 (1999), 1059-1066. J. I E. Urbas, J. Differ. Geom. 33 (1991),91-125. R. S. Hamilton, J. Differ. Geom . 24 (1986), 153-179. B. Chow, 1.P. Liou and D.H. Tsai, Comm. Anal. Geom. 3 (1996), 75-94.

A Symplectic Transformation and its Applications * Mei-Yue Jiang Department of Mathematics Peking University, Beijing, 100871, China R n is presented. Some applications are given, which include periodic solutions of Hamiltonian system and construction of symplectic embeddings from (R 2n , wo) to some symplectic manifolds.

Abstract A simple and elementary symplectic transformation on Tn

X

1. A Symplectic Transformation Let Tn X R n be the cotangent bundle of the torus Tn and >. = E~ Yidxi be the Loiuville form, where x = (Xl,· · ·, Xn) E Tn, each Xi is I-periodic, and Y = (YI, .. . , Yn) ERn. We consider the following symplectic form on Tn X R n,

w

= d>' + n = Wo + n,

where Wo = E~ dYi 1\ dXi is the standard symplectic form on Tn X R n, and n is a closed 2-form on Tn. Such w is called twisted symplectic form. The physical meaning of n is a magnetic filed on the torus. The Hamiltonian systems with this twisted symplectic form on the cotangent bundle was studied in the early eighties by Novikov. Some important classical Hamiltonian systems arising from physics can be written in this form, see [20]. In this article, we survey some results obtained by the author related to Tn X Rn with the twisted symplectic form w, which include the periodic solutions on a given energy surface, construction of symplectic embbedings from (R 2n , wo) to some symplectic manifolds and periodic solutions of a class of superquadratic Hamiltonian systems. All these results are based on following transformation, a simple linear algebra fact . PROPOSITION 1.1. Let n = aijdxi 1\ dXj be a constant and non-zero

E;:;

·supported by NNSF of China and Ministry of Education of China

175

Mei- Yue Jiang

176

2-form on Tn andw = wo+n. Then there exists a diffeomorphism ¢ ofTnxRn of the following form

¢ : (X, Y) ---t (x, y) , x

= X + B . Y,

y

=C .Y

where Band Care n x n matrices and det( C) -:j; 0 such that r

¢*(w)

n

= LdYit\dYi+r+ L>ijdXit\dXj +

L

dijdYit\dXi ,

(1.1)

i~2r+1

I

where 2r = rank(A), and (d ij ) is a constant matrix. Proof: The matrix A = (aij) is non-zero and anti-symmetric, so the rank of A is an even number 2r . Let be the r x r identity matrix and

Ir

J =

(0 -Ir) . Ir

,

0

For a matrix B = (b ij ), we write Bdx t\ dy tions show that

¢*(w) = AdX t\ dX

= L:ij bijdxi t\ dXj.

+ (2A · B + C)dY t\ dX + (Bt

. A- B

Simple calcula-

+ Bt . C)dY t\ dY,

where the upper t denotes the transpose of a matrix. The matrices B and C can be obtained as follows . Let B be an invertible n x n matrix such that

Bt . A . B

= ( ~,J2r' ~)

and the matrix C be determined by

Bt . C = ( - ~J2Tl 0,

0 ) . I n - 2r

Then it is easy to see that Band C satisfy (1.1) . As a corollary, we have following result, see also [6]. COROLLARY 1.2. Let n = L:~.j aijdxi t\ dXj be a constant symplectic form on T2n and w = Wo + n. Then there exists a diffeomorphism ¢ of T 2n X R 2n of the following form

¢ : (X, Y) ---t (x, y), x

= X + B . Y,

y

=C .Y

where Band Care 2n x 2n matrices and det( C) -:j; 0 such that 2n n ¢*(w) = L dYi t\ dYi+n + L aijdXi t\ dXj = WI . I

(1.2)

I

On the right hands of (1.1) and (1.2), the first term L:~ dYi t\ dYi+r and L:~ dYi t\ dYi+n are indepedent of other variables, this point is crucial for the applications later.

A symplectic transformation and its applications

177

2. Symplectic Capacity and Periodic Solutions In this section, we study the existence of periodic solutions of Hamiltonian systems on a given energy hypersurface. In order to state our results, we need a symplectic capacity which is defined by Hofer and Zehnder as follows. Let (N,w) be a symplectic manifold, we denote ll(N,w) the set offunctions H on N satisfying (1) there is an open nonempty set U C N such that H(x) = 0 for x E U; (2) there are compact set K C int(N) and positive number m(H) such that H(x) = m(H) for x E N \ K; (3) 0 ~ H(x) ~ M(H) for x E N. For a function H on N, let XH(x) be the Hamiltonian vector field defined by w(XH ,·) = dH( ·). DEFINITION 2.1. HE ll(N,w) is called admissible if :i;

= XH(x)

has no nonconstant T-periodic solution for 0

" + O. Then for any open subset U of T*Tn with compact closure, we have c(U, w) < +00. This theorem has some consequences for periodic solutions of autonomous Hamiltonian systems (HS) i = XH(z), Z = (x, y) on a given hypersurface E = {(x,y)IH(x,y) = c}. E is called regular if

E

= {(x, y) E Tn

X

Rn, H(x, y)

= c}, dH(x, y) :f 0

for

(x, y) E E.

Note that the existence of periodic solution of (HS) on E depends only on the hypersurface E. THEOREM 2.3. Let 0 be as above and E be a compact hypersurface of contact type of (Tn X R n, d>" + 0) . Then there exists a periodic solution of (HS) on E which is contractible in Tn X Rn provided E is regular. Let H be a smooth Hamiltonian function on Tn xRn such that for all h,Eh = ((x,y) E Tn X Rn,H(x,y) = h} is compact. Set

C

= {h E R,dH(x,y):f 0

for

(x,y),H(x,y)

= h} .

178

Mei- Yue Jiang

Then for almost all h E C, Hamiltonian equation (HS) has a periodic solution on 'Eh which is contractible in Tn X R n. A compact hypersurface 'E is of contact type, if there is a vector field ~ defined in a neighborhood of 'E which is transversal to 'E and satisfies L{w = w. Remark 2.4. Theorem 2.2 has been known to experts for some time. IT the cohomology class of n is zero, it is proved in [10]. In this case, the periodic solution obtained may not be contractible in Tn X Rn. IT the cohomology class of w is rational, it can be deduced from a result in [11]. The case that n is nondegenerate, hence n must be even, is proved in [6]. The formulations as stated is proved in [13], see also [8]. For a generalization, see [19]. Remark 2.5. There are some results for n = 2, see [7], [21] and [25]. In particular, following result is proved by Arnold and Kozlov. Let n = a(xI, X2)dxIl\dx2, a(xI, X2) =I 0 for any (Xl, X2) E T2. Then for h > > 1, there are three periodic solutions of (H S) satisfying H(x, y) = 2:; IYil 2 + V(x) = h. The proof relies on the well known symplectic fixed point theorem due to Conley-Zehnder, see [15]. Proof of Theorem 2.2: Since n a closed, non exact 2-form on Tn, by the de Rham theorem, there are constants Cij, 1 ~ i,j ~ n and I-form 0: on Tn such that n = Cijdxi 1\ dXj + do:

t

L i,j

iTn aij(x)dx,Cij =I 0 for some i,j, since {dXi I\dxjh from (R 4 , to (~x R2,wEBwo). THEOREM 3.2. Let w be a constant symplectic form on T2n. Then the7 a symplectic embedding if> from (R 2n+2m,wo) into (T2n X R2m,wEBwo) . The proof of theorems is quite elementary and the symplectic embeddi can be constructed explicitly. It is based on the proposition 1.1 and the that the universal covering of ~ is R 2 and that of T 2n is R 2n. For any two ( forms WI and W2 on ~, by Moser's theorem, there are diffeomorphism if> ( and constant c such that if>*(wd = c· W2, see [9]. Thus we can fix a symple form w as follows. If ~ = T2, then we set w = dXl 1\ dX2, if the genus ( 9 2: 2, then set w be the area form of the metric of curvature -1. In follov we use Xi, Xi to denote the coordinates of ~ or T 2 n, and Yi, Yi denote tha R2n. Let U(a, b) = (a, b) x R, X E (a, b), y E R, Wo = dy 1\ dx . The follov simple geometric fact will be used. LEMMA 3.3. Let U be a contractible domain ofR2 such that I fuwol =-1 then (U, wo) is symplectomorphic to (R 2 , wo) . Proof of theorem 3.1: By proposition 1.1, there is a symplectomorpsim

if> : (T2

X

R2, wEB wo) -+ (T2 x R2, dYl

1\

dXl

+ dY2 1\ dX2 + dXl

1\

dX2)'

xR2 ),

On the open subset U(O, 1) x U(O, 1) of (T2, the twisted symplectic £ is symplectomorphic to Wo EBwo . Thus the restriction of if>-l to U(O, 1) x U(f produces a symplectic embedding from (U(O, I) x U(O, 1), Wo EB wo) to (1 R2, wEB wo). This proves theorem 3.1 for the case ~ = T2 by lemma 3.3. Now we assume the genus of ~ g 2: 2. Let w be the area form of the mE on ~ with curvature -1. The universal covering of ~ is the hyperbolic Sl (H2, ds 2 ), where

H2

= {(X 1, X)2

E R2 ,X 2

> O} ,s d 2 -_ dXf X2 + dX? 2

Let P be the covering map, then P*(w) = dX';!X For 8 > 0, cons 2 if>l: U(-8,8) x U(-8,8) -+ H2 x R2, (Xl,X2,Yl,Y2) -+ (X l ,X2,Yi,Y2), 2 •

Xl = -e(Yl-:l:2) (Xl Y 1 = Yl,

+ Y2), X 2 =

e(Yl-:l:2),

Y2 = -Y2,

A symplectic transformation and its applications

181

then

4>i(P*(w) ffi wo) = W2 = dXI 1\ dX2

+ dYI 1\ dXI + dY2 1\ dX2·

For 0, set

Uo = {(x,y) E R2, Ixi < 6, IxYI < 6}. Let 4>2 : U ( -6, 2(XI,X2,YI,Y2) = (P(X I (X,y),X 2(x,Y)),YI,-Y2). It can be shown that for small 6 > 0, 4>2 is a symplectic embedding from (U(-6,6) x Uo,W2) to (1: x R2,W ffiwo) . Now we fix a such 6 and show that (R\wo) and (U(-3 : (R4n,wo) -+ (T2n X R2n,wd. Consider the set

{(X,Y) E T 2n

X

R2n,0

< Xi < 1,li = 1, ··· ,2n},

on this set we have WI = L~n d(Y + AX) 1\ dX and (X, Y) -+ (X, Y + AX) is a diffeomorphism, thus it is symplectomorphic to n products of U(O, 1) with symplectic form Wo = L~n dY; 1\ dX i , which is symplectomorphic to (R4n,wo). This proves theorem 3.2 if n = m. The remaining part of theorem 3.2 can be proved as follows . We assume m = 1, n = 2, the other case is similar. It is enough to show that for small 6, there is a symplectic embedding from symplectic manifold (U(0,6) x U(0,6) x U(O, 6), Wo ffi Wo ffi wo) to (T4 x R2, W ffi wo) . This will be completed by two steps. Let P : R4 -+ T4 be the covering map, we first find a symplectic embedding IJ! from this manifold to (R4 x U(0,6),wo ffiwo). Then choosing a linear symplectomorphism S : (R 6 , wo) -+ (R4 x R 2 , W ffi wo) properly, we can show that (P, I dR 2 )oSolJ! is a needed symplectic embedding to (T4 XR2, wffiwo). The map IJ! can be given explicitly by following way. Take a diffeomorphism ! : R -+ (0,6), set 4>4: (U(0,6) x U(0,6),wo ffiwo) -+ (R2 X U(0,6),wo ffiwo),

4>4(Xl,Yl,X2,Y2)

= (-Y2 -Xl,Yl,!(Y2),

+ X2) !'(Y2) ).

YI

182

Mei- Yue Jiang

Let 111 = (Id U (O ,o),¢4) 0 (¢4,Id u (o,o)). The linear transformation S is given by following lemma. Let {el ,· ·· ,e2n} be the standard orthogonal basis of R2n . Recall that a symplectic basis for w = L bijdxi 1\ dXj is a basis {II,· .. , hn} such that w(j2i-I,hi) = 1 = -w(j2i,hi-d , i = l,···,n,

w(fi,!i)=O for other i,j. LEMMA 3.4. Let w = L bijdxi 1\ dXj be a constant symplectic form on R2n. Then after a permutation of the basis {eI,· · ·,e2n}, still denote it by {el, . .. , e2n}, there is a symplectic basis {II, ·· ·, hn} for w such that Ii E span{el,··· ,ed· This lemma can be proved by induction on n, see [13] . Let (II, · .. , hn) = T· (eI, . . . ,e2n) and let S be the linear transformation defined by

Then S·w = Wo = L~ dY2i-1 1\ dY2i and S considered as made up of 2 x 2 blocks, is a block of upper triangular matrix. For w EEl Wo on R 6 , we have a such matrix S = (Sij) . Then elementary computations show that for 8 small, (P,Id R 2) 0 S 0 111 U(O,8) x U(O,8) x U(O,8) -+ T4 x R2 is an embedding satisfying This completes the proof of theorem 3.2. As a consequence of above construction, we can obtain some estimates of another symplectic capacity, the Gromov's capacity, which is defined by cI(M,w)

= SUp{11T2 , 3a

symplectic embedding

¢ : (B 2n (r) , wo) -+ (M,w)} .

THEOREM 3.5. Let w be a constant symplectic form on T2n , and (~, r) be a closed 2-dimensional symplectic manifold of genus g ~ 1. Then there is a constant C > 0 such that for R > > 1, the following inequalities hold: (1) cI(T 2n X B 2(R) , w EEl wo) ~ C · Rn~l . (2) CI (~ x B2(R) , rEEl wo) ~ C· log(R) if the genus of ~ g ~ 2. REMARK 3.6. It is easy to see from comparison of the volume that the exponent n!l in (1) is optimal. For the standard 2n-torus (T2n , wo), Wo =

L~ dX2 i-1 1\ dX2i, we can take C = (~)n~l . For any 2-dimensional symplectic manifold (~, r) with 1IE rl = 1r R 2, since for any r < R, there is a symplectic embedding from (B2(r), wo) to (~, r), so

A symplectic transformation and its applications

183

monotonicity of cl(M,w) implies

COROLLARY 3.7. Let (E, r) be 2-dimensional symplectic manifold, then if

4. Periodic Solutions of Nonautonomous Hamiltonian Systems In this section, we consider periodic solutions of Hamiltonian systems on Tn X Rn for time dependent Hamiltonian H which is periodic in the variable t with period 1 with the twisted symplectic form. We assume that n = " .. aiJ·dxi 1\ L.-i'l. ,) dx j is a constant symplectic form . Thus the Hamiltonian system can be written as (4.1) Ax - iJ = Hz(t , x, y). x = Hy(t, x, y), It will be convenient to consider (4.1) as a Hamiltonian system on R 2n with the Hamiltonian H being periodic in variables Xi, i = 1,2" . . ,n with period 1 later. Since Conley-Zehnder's solution of the Arnold conjecture on torus, see [4], there have been many works on existence of periodic solutions for spatially periodic Hamiltonian syetems, either on torus or on Tn X R n with A = 0, we refer to [2], [3] and the references theirin. Our aim here is to prove following THEOREM 4.1. Let H E C1(R X T 2n X R2n) satisfy (H1) H(t, X, y) = H(t + 1, X, y), H(t, X, y) = H(t, -X, -y); (H2) there are constants IL > 2, Ro > 0 such that

Hy(t,x , y)· Y 2: ILH(t,x,y) > 0

for

(t,x) E R

X

T 2n , Iyl 2: Ro;

(H3) IHzl ~ M for some constant M > 0,(H4) H 2: 0 and H(t, X, y) = o(lxl 2 + IYI2) near (0,0),(H5) there are constants Cl, C2 > 0 such that IdH(t, X, y)1

~ Cl

+ c2Hy . Y

for

any

(t, X, y) .

Then there is a sequence of 1-periodic solutions (Xj(t), yj(t)) of (4 .1) which are contractible in T 2n X R 2n . There are many results as above for variational problem, i.e. evenness and a kind of superquadraticity ensure infinitely many solutions. The case A = 0 has been considered in [2]. For the case A = 0, one can expect existence of noncontractible periodic solutions in other homotopy classes, however, simple examples show that this is impossible if A i- O. The reason for this phenomenon

Mei- ¥ue Jiang

184

is that (4.1) is equivalent to a Hamiltonian system on T 2n X R 2n with a spliting symplectic form by proposition 1.1. This enables us to use some existences results for periodic solutions of Hamiltonian system on R 2n with slight modifications . PROOF OF THEOREM 4.1: By proposition 1.1, (4.1) is equivalent to A~

= K~(t,~, 1]),

J17

= K.,,(t,~, 1])

(4.2)

with K(t,~, 1]) = H(t, ~ + B1], C1]), and J is the standard symplectic matrix on R 2n. It is easy to see that the new Hamiltonian K satisfies same conditions as in theorem 4.1 if (x,y) is replaced by (~,1]). We will show that there is a sequence of I-periodic solutions (~i,1]i) of (4.2) such that

where I(~,1]) =

(II.

10

(2(A~,~)

1

+ 2(J17,1]) -

K(t,~,1]))dt.

(4.3)

For simplicity, we assume A = J, the standard symplectic matrix on R 2n and the Hamiltonian H satisfies (H6) IdH(t,x,y)1 ::; C3 + c41ylS for some constants C3,C4 and s. This condition can be removed by a trunction procedure as in [23). With this condition, the functional I in (4.3) is C 1 and even on the space (~, 1]) E X = H t (Sl , R 2n) X H t (Sl , R 2n). The proof given below is a slight modification of that in [23], therefore it will be sketchy. Let X = X_ EB Xo EB X+, where X_, X+ and Xo are the negative, positive and zero space of the quadratic form

We know Xo = R 2n X R2n . (HI) implies that I is well defined on X_ EB (T2n X R2n) EB X+. LEMMA 4.2. The functional I(~, 1]) satisfies the (P.S) condition on X_ (B (T2n X R2n) EB X+ . Proof: Let (~I' 1]1) be a (PS) sequence, it is easy to show that lI~dl is bounded modulo the translation of the integers. With this fact in hand, the boundedness of II1]dl and a convergent subsequence can be proved as in [1). Now we consider I as a functional on X, (P.S) condition fails, however, the deformation lemma holds for I on X because of above lemma, thus critical point theorems can be applied.

A symplectic transformation and its applications

Let S be the family of sets A w.r.t. O. For A E S, let

c

185

X \ {O} such that A is closed and symmetric

')'(A) = min{jI3¢ E C(A,Rj \ {O}),¢(-x) = -¢(x),x E A}, this is the Z2 index. For properties of this index, we refer to [24]. Let

By (H2), there are constants

C5, C6

such that

hence we get (4.4) Since dimVk n X+ rk > 0 such that

= 2kn and J.L >

2, (4.4) and (H4) imply that there is an

I(~,"1) ::; 0 for z E Vk , IIzil ~ rk . 2 As H(t, x, y) = o(lxl + IYI2) near (0,0), we have K(t,~, "1) (0,0), therefore for small p > 0,

(4.5)

= o(I~12 + 1"112) near (4.6)

for some constant a . Let B(r) be the closed ball of radius r in X centered at O. Set Dk = B(rk) n Vk . Let P_ be the projection from X to X_ and let G k be the class of maps h E C (D k, X) with following properties;

(1) h( -x) = -h(x); (2) h(z) = z for z E 8D k ; (3) P_h(z) = a(z)L + g(z), where 9 is compact and a E C(Dk' [1, aD, a depending on h. Finally for j EN, set

rj

= {h(Dk \ Y)lk ~ j, hE G k , YES, Cj

and ')'(Y)::; 2nk - 2nj},

= infBEr;suPzEBI(z).

The conclusion that {Cj} is an unbounded sequence of critical values of I follows from the standard deformation argument. This completes the proof of theorem 4.1.

186

Mei- Yue Jiang

Remark 4.3. In order to use the Z2 mountain pass theorem to obtain an unbounded sequence of critical values for an even functional, one usually need to verify that (4.6) holds and (4.5) holds for X_ EEl Xo EEl Ek, where Ek is an arbitrary sequence of finite dimension subspaces of X+ such that dim(Ek ) -t +00. However, the same conclusion hold if for one such sequence, (4.5) holds, see [5] (corollary 7.23) if dim(X_ EElXo) < 00. In our case, dim(X_ EElXo) = 00, this can be treated by a Galerkin approximation. Remark 4.4. If we assume the growth condition (H6), then (H5) can be dropped. (H5) is only needed in the trunction. If we consider autonomous Hamiltonian system, that is H is independent of time t, then we can use SI symmetry instead of Z2 symmetry. In this case, conditions (H5) and symmetric condition are not needed.

References [1] A. Bahri and H. Berestycki, Forced vibrations for superquadratic Hamiltonian systems, Acta Math., 152 (1984), 143-197. [2] T. Bartsch and Z. Q. Wang, Periodic solutions of spatially periodic, even Hamiltonian systems, J . Diff. Equa., 135 (1997), 103-128. [3] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhiiuser, Boston, 1993. [4] C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture by V. I. Arnold, Invent. Math., 73 (1983), 33-49. [5] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press, 1993. [6J V. 1. Ginzburg, On closed trajectories of a charge in a magnetic field. An application of symplectic geometry, in Contact and Symplectic Geometry, ( C. B. Thomas ed.), pp131-148, Cambridge University Press, 1996. [7] V. L. Ginzburg, On the existence and non-existence of closed trajectories for some Hamiltonian flows . Math. Z. (1996), 223: 397-409. [8] V. L. Ginzburg and E. Kerman, Periodic orbits in magnetic fields in dimension great than two, preprint, 1999. [9J Hofer H. and Zehnder E. Symplectic Invariants and Hamiltonian Dynamics. Boston Basel, Birkhiiuser, 1994. [10J M.-Y. Jiang, Hofer-Zehnder symplectic capacity for 2-dimensional manifolds. Proc. Roy. Soc. Edinburgh (1993), 123A: 945-950.

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[11] M.-Y. Jiang, Periodic solutions of Hamiltonian systems on hypersurfaces in a torus, Manuscripta Math., 85(1994), 307-321. [12] M.-Y. Jiang, Symplectic embeddings from (R2n,wo) to some symplectic manifolds, Proc. Roy. Soc. Edinburgh, 129A, 1999( to appear) . [13] M.-Y. Jiang, Periodic motions of particles on torus with a magnetic field. Research Report No.21, Institute of Mathematics, Peking University, 1997. [14] M.-Y. Jiang, Periodic solutions of a class of Hamiltonian systems on the cotangent bundle of torus, Research Report No.27, Institute of Mathematics, Peking University, 1998. [15] V. V. Kozlov, Variational calculus in the large and classical mechanics. Russian Math. Surveys (1985), 40: 37-71. [16] F . Lalonde, Isotopy of symplectic ball, Gromov's radius and the structure of ruled symplectic 4-manifolds. Math. Ann. 300 (1994), 273-296. [17] F . Lalonde and D. Mc Duff, The geometry of symplectic energy. A nn. of Math . 141 (1995),349-371. [18] D. Mc Duff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs: Clarendon Press, 1995. [19] G. C. Lu, The Weinstein conjecture on some symplectic manifolds containing the holomorphic spheres, Kyushu J. of Math., 52 (1998), 331-351. [20] S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory. Russian Math. Surveys 37(1982), 1-56. [21] S. P. Novikov and P. G. Grinevich, Nonselfintersecting magnetic orbits on the plane. Proof of the overthrowing of cycles principle. Amer. Math. Soc. Transl. (2) 170 (1995), 59-82. [22] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math ., 31 (1978), 157-184. [23] P. H. Rabinowitz, Periodic solutions of large norm of Hamiltonian systems, J. Diff. Equa., 50 (1983),33-48. [24] P. H. Rabinowitz, Minimax Methods in Critical Point theory with Applications to Differential Equations, CBMS, Regional Conf. Ser. in Math., 65, Amer. Math. Soc., Providence, RI, 1986. [25] I. A. Taimanov, Closed extremals on two-dimensional manifolds. Russian Math. Surveys (1992), 47: 163-211.

Positive Solution to p-Laplacian Type Scalar Field Equation in RN with Nonlinearity Asymptotic to u p - l at Infinity 1 Gongbao L£2, Lina Wu and Huan-Song Zhou Young Scientist Laboratory of Mathematical Physics Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences P .O.Box 71010, Wuhan 430071, P .R. China Email: ligblilwipm . whcnc.ac . cnandhszhou~wipm.whcnc.ac.cn

Abstract

We consider the following elliptic problem:

-div(lV'ulp-2V'u) + mlul P - 2 u { u E W1,P(R N ), N > p > 1,

= f(x, u)

where m > 0, ,~f:.'..t tends to a positive constant as u -+ +00. In this case, f(x, u) does not satisfy the following Ambrosetti-Rabinowitz type condition, that is, for some () > 0,

Ai"

o :5 F(x, u) =

0

1 () f(x, u)u for all (x, u) E R N x R, f(x, s )ds :5 p +

which is important in applying Mountain Pass Theorem. By a variant version of Mountain Pass Theorem, we prove that there exists a positive solution to the present problem . Furthermore, if f(x, u) == f(u), the existence of a ground state to the above problem is also proved by using artificial constraint method.

1

Introduction

In this paper, we deal with the existence of a positive solution to the following non-autonomous quasilinear scalar field equation: -div(lV'ulp-2V'u) + mlul p - 2 u { u E W1,P(R N ), N > p > 1, 1 Partially Supported by NSFC 2Partially Supported by Academy of Finland

188

= f(x,u)

in RN,

(1.1)

Positive solution to p-Laplacian type scalar field

189

where m > 0 is a constant. The conditions imposed on f(x, t) are the following: (Cl): f: RN x R ~ R satisfies the Caratheodory conditions, i.e. for a.e. x E RN, f(x, t) is continuous in t E R and for t E R, f(x, t) is Lebesque measurable with respect to x ERN; f(x, t) ~ 0, "It ~ 0, x ERN; f(x, t) == 0, \:Ix E RN, t < O. (C2): lim ~1::P = 0 uniformly in x ERN . t--+o

(C3) : There is a constant f E (0, +00) such that lim ~1::~) = f uniformly t--++oo in x ERN. (C4) : For each x E RN, ~1::P is non decreasing with respect to t > O. RN

(C5) : There is a function f(t) E C(R) with f(x, t) ~ f(t) for any (x, t) E N : f(x, t) > !(t)} > 0 for any t > 0 such that X Rand mes{x E R lim f(x, t) = f(t) uniformly in bounded t. 1"'1--++00 (C6):

!

E Cl(R) and

(p - 1)!(t) < f'(t)t for all t > O.

Definition We say that u E W1,P(R N ) is a nontrivial (weak) solution to problem (1.1) if u t 0 and satisfies

r {1Y'ulp-2Y'u.Y'cp+mluIP-2ucp}dx = iRN r f(x, u)cpdx iRN

for all cp E W1,P(R N ).

Moreover, we say that u E W1,P(R N ) is a positive solution to problem (1.1) if u is a nontrivial weak solution of (1.1) and u(x) > 0 a.e. in RN . Clearly, solutions to (1.1) correspond to critical points of the following energy functional defined on W1,P(R N ), I(u) =

!

P

r (lY'uIP+mluIP)dx- iRN r F(x,u)dx, iRN

where F(x,u) =

1"

f(x,s)ds .

0

(1.2) The very useful tools to get critical points of I(u) are the famous "Mountain Pass Theorem" proposed by Ambrosetti and Rabinowtiz in their paper [1] and the constraint minimization. Combining Mountain Pass Theorem and Sobolev imbedding theorem, people have attained a lot of achievements in studying the existence of solutions of nonlinear Dirichlet problem in bounded domain n (see, e.g. [13]). Problem (1.1) occurs in many applications and there is an enormous literature concerning the existence of nontrivial solutions for problems similar to (1.1), or its bounded-domain counterpart under various hypotheses on f(x, t), see, e.g. [1] [2] [4] [11] [12] [18] [20] and the references therein.

Gongbao Li, Lina Wu and Huan-Song Zhou

190

Semi-linear elliptic equations in R N , e.g. problem (1.1) with p = 2, were motivated in particular by the search of certain kinds of solitary waves in nonlinear equations of the Klein-Gordon or Schrodinger type (see, e.g. [2) [3) [4)). Evidently, a contrast between semi-linear elliptic boundary value problems on a bounded domain and on RN is the apparent lack of compactness of Sobolev imbedding in treating latter. A partial reason for the lack of compactness of the Sobolev imbedding is the invariance of RN under the translations, rotations and dilations. To overcome this difficulty, people have attempted various methods, e.g. by working with some appropriate constraint in order to have some compactness. For example, to the following scalar field equation:

-D.u = f(x,u), { u E HI (RN),

x ERN, u"t. 0,

(1.3)

when f(x, u) is spherical symmetrical (i.e. f(x, u) = f(lxl, u)) or f(x, u) is autonomous (i.e. f(x,u) == f(u), "Ix ERN), we can consider problem (1.3) in the radial symmetrical Sobolev Space H~(RN) = {u E HI(R N ): u(x) = u(lxl), "Ix ERN}.

By the compactness of the imbedding H:(R N ) Y Lq(RN)(2 < q < ~~2 if N > 2), the existence of solutions to problem (1.3) in the Sobolev space H: (RN) were considered by Strauss[4]' Berestycki-Lions[2)[3). However, these methods does not work for general f(x, t). For this purpose, the "ConcentrationCompactness Principle" was proposed by P-L.Lions, in which some necessary and sufficient conditions for the compactness of minimizing sequence of a functional in unbounded domain are given. Based on this principle, the variational elliptic problems in RN were studied widely, see [6).....,[10) and the references therein. In [12], Yang and Zhu studied problem (1.1) under the basic assumptions similar to (C1) (C2) (C4)-(C6) and f(x, t) is subcritical, furthermore, they require as usual the following technical condition proposed by Ambrosetti and Rabinowitz in [1], that is, for some () > 0,

(U

1

0::; F(x,u)~ 10 f(x,s)ds::; p+(}f(x,u)u, V(x,u) ERN

X

R.

(1.4)

This condition implies that for some C > 0, F(x, u) 2: Clul P+9 for u > large, which makes the functional J satisfy the "mountain pass structure" and (1.4) also ensures that every (PS)c sequence {un} is bounded in HI (R N ). By applying the concentration-compactness principle, they proved that J(u) satisfies (PS)c condition for < c < Joo (see below for the definition of J oo ),

°

°

Positive solution to p- Laplacian type scalar field

191

hence the existence of a nontrivial solution of (1.1) is obtained by Mountain Pass Theorem. However, (C3) implies that !(x, t) is asymptotically "linear" with respect to u p - 1 at +00, and (1.4) can not be true. The study of the existence of positive solution to problem (1.1) under the condition (C3) is related to seeking special solutions of Maxwell's equations (see, e.g. [14] [15]). In this case, problem (1.1) with p = 2 have been studied in the spherical symmetric Sobolev space by Stuart and Zhou in [20] [21] [28]. Without assuming the symmetry on !(x, t), Jeanjean in his paper [18] proved that there is a positive solution to problem (1.1) with p = 2 under more or less the same assumptions as (C1) - (C4), moreover, he requires that !(x, t) is I-periodic in x ERN, which makes problem (1.1) invariant under suitable translations. But, he can't assure whether this solution achieved the mountain pass level of the energy functional / or not. Recently, Li and Zhou in [16] proved the existence of a positive solution to problem (1.1) for p = 2 under more general assumptions on !(x, t) as (C1)-(C6) with p = 2 and proved that this solution can achieve the mountain pass level of /. The aim of this paper is to generalize the main results of [16] for general p > 1. To state our main results, we first define the problem at infinity associated with problem (1.1) as follows: -div(lV'uIP:"'2V'u) { u E W1,P(R N ),

+ mlul p - 2 u =

f(u), x ERN,

(1.5)

and for any u E W1,P(R N ), define /OO(u) = where P(u)

~

p

= IoU f(s)ds.

[

iRN

(lV'uI P + mlulP)dx - [

iRN

P(u)dx,

(1.6)

Clearly, /00 E C1(W1,P(R N ), R). Denote

A = {u E W1,P(R N ) : (/00' (u),u)

= O,u ~ O},

(1.7)

where (', -) is the usual dual paring between W1,P(R N ) and its dual space W-1,p' (RN) (p' = f-r), [00' is the Frechet derivative of / 00 . It can be sho~n that A :f. 0 (see Lemma 3.1 in section 3 below). We define now the followmg minimization problem: JOO

= inf{IOO(u) :

u E A}.

(1.8)

We recall that if a solution of (1.5) achieves Joo in (1.8), it is called a "ground state" for (1.5) . Our main results are as follows:

Gongbao Li, Lina Wu and Huan-Song Zhou

192

Theorem 1.1. Let e E (m, +00) and conditions (C1)-(C6) hold, then Joo > 0 and it is achieved by some Uo E Wl'P(RN). Moreover, uo(x) > 0 a.e. in RN, which is a ground state for problem (1.5). Theorem 1.2. Suppose that conditions (C1)-(C6) hold, then problem (1.1) has a positive solution if E (m, +00) and there is no positive solution to (1.1) if e:s; m .

e

We mention that, to our best knowledge, the above results have not been seen elsewhere. To prove our main results, the difficulties are caused by the lose of condition (1.4) and the compactness of Sobolev imbedding in RN Since lim ~1~~} =

t-++oo

e < +00,

(1.4) is not satisfied, even in the case of p = 2, Joo > 0 is no longer apparent and not every minimizing sequence of Joo is apparently bounded as in the case of lex, t) satisfying (1.4). To prove Theorem 1.1, we use the Ekeland's variational principle on Finsler manifold (see Lemma 2.6 in Section 2, or [16]) to get a special minimizing sequence {un} of Joo in W1 ,P(R N ) with 11100 (un)llw-l.pl (RN) ~ 0, then as in [16] we use the concentration-compactness principle to show that Joo > 0 and Joo is achieved. To prove Theorem 1.2, as in [16] we find that the positive solution Uo of (1.5) obtained by Theorem 1.1 satisfies 1 (Uo ( I)) --+ - 00 as t --+ + 00 and we can construct the mountain pass level as follows c = inf max leu) , (1.9) 1

-yEr uE-y

where

r

=

b

E C([O, 1], Wl 'P(RN)) : 'Y(O) = 0, 'Y(1) = uo( ~)} with some

to to > 0 large enough. Then by the Mountain Pass Theorem without (PS) condition as in [19], there is a (PS)c sequence of 1 such that

To prove that {un} is bounded in W1,P(R N ) is a typical difficulty in the case of (C3) for the lack of (1.4). For this step, we follow basically the processes as in [16], of course, there are some technical difficulties while we work in the Banach space Wl ,P(R N ) with general p > 1 instead of the Hilbert space Hl(RN) = Wl,2(RN) . To overcome the difficulty of the lack of compactness of Sobolev imbedding from W1,P(R N ) to U(RN), we use a different approach from that of [16] . Instead of using the complete long procedure of concentration-compactness principle, we find that by comparing the energy level of two different critical points, some desirable results can be obtained. More precisely, we prove that

Positive solution to p-Laplacian type scalar field

193

°

if {Uk} is a bounded (PS)c sequence of [ with c > and Uk ~ Uo weakly in WI,P(R N ), then either Uo E WI ,P(R N ) \ {o} with I'(uo) = 0, or c ~ Joo. This is similar to Proposition 3.1 in [17), but we don't assume that f(x,') E CI as [17] did. In fact, the method of [17] seems only applicable to the case where m f(x, u) = L Qi(x)lul qi - 2 U (qi > p, Qi(X) E LOO(RN )). As we can show by i=1

using Pohozaev identity for p-Laplacian type equation (see Lemma 2.1 below) and the definition of c in (1.9) that c < J oo , then Theorem 1.2 is proved. Clearly, our approach here is much simpler than that of [16] even if p = 2. We end this section by giving some notations. Throughout this paper, we denote by C a universal positive constant unless specified and, define the norm of u E WI,P(R N ) by

lIuli ~ Ilullwl.P(RN) = (IN (IVuI P + m1uIP)dX)

l. P ,

and the norm of u E £p(RN)(1 < p < +00) by

We use" -t" (" -''') to indicate the strong (weak) convergence in corresponding function spaces.

2

Preliminary results

In this section, we give some preliminary results which will be used frequently in the following text. By (Cl), it is clear that f(x, t)

= f(x, t+),

V(x, t) E RN

X

R.

(2.1)

It follows from (Cl) - (C5) that for any c > 0, q E (p,p*), (N > p), there exists CE > such that for all x ERN, t ~ 0,

°

:!p

f(x, t) :::; ctp F(x, t) :::;

I

+ CEt q - l ,

clW + CEIW,

f(t):::; ctp F(t):::;

I

+ CEt q - l ,

p*

=

(2.2)

clW + CEIW,

(2.3)

f(x, t) :::; UP-I, f(t):::; Up-I.

(2.2')

By (C4) (C5) we see that F(x, t) < ~ f(x, t)t -p

for all x E RN, t

~ 0,

F(t):::;

~p f(t)t

for t

~ 0.

(2.4)

Gongbao Li, Lina Wu and Huan-Song Zhou

194

Combining (C2) (C3) and (C5), we have that lim f(t)

t-tO t p - 1

= 0,

lim

f(t)

t-t+oo tp-l

=t.

(2.5)

FUrthermore, (C6) implies that

f(t) tp -

is strictly increasing in t

1

> 0 and F(t) ~

Jot

f(s)ds
0 and by J+oo i'fdR R--t+oo 18BR 1 oo that Jo+ (J8BR lV'uoIPdS)dR = +00 which is impossible, since

= +00

we see

Therefore

P-Nj -

P

and Since

N -~p(uo)

B~

j RN

n P-Nj uog(uo)dx ---+ --

P

~

uog(uo)dx,

P-Nj G(uo)dx + uog(uo)dx = P RN

= g(uo)

o.

and by integrating by parts we have

Lemma 2.2. Let {Pn} C Ll(RN) be a bounded sequence and Pn ~ 0, then there exists a subsequence, still denoted by {Pn}, such that one of the following two possibilities occurs: (i) (Vanishing): lim sup J, +B Pn(x)dx = 0 for all 0 < R < +00. n--t+oo yERN y R (ii) (Nonvanishing): There exist Q > 0, R < +00 and {yn} C RN such that lim n--t+oo

r

lYn+BR

Pn(x)dx

~ Q > O.

The proof of this lemma is trivial. Lemma 2.3. (Vanishing Lemma, Lions [9]) Let 1 < P ~ +00, 1 ~ q < +00 with q:f:. :!p if P < N . Assume that {un} is bounded in U(RN), {V'u n } is bounded in £p(RN) and sup J, +B lunlqdx ~ 0 for some R > 0, then

yERN y

R

196

Un ~

Gongbao Li, Lina Wu and Huan-Song Zhou

°

in U>:(RN) for any a between q and NNp .# -p

Lemma 2.4. For I defined by (1.2) , if {un} C Wl ,P(R N ) satisfies (1'(un),U n ) ~ 0. Then, for any t > 0, by extracting a suitable subsequence, we have

1(tun )

~

1 + tP pn

- - + 1(u n ).

°

In particular, if 'In ~ 1, (I' (un), un) = then 1(tun ) ~ 1(u n ) for any t > 0. Proof. This Lemma is proved in [20] for P = 2 and in [27] for general p> 1.# Finally, we give several results for the minimization problem (1.8) . For this purpose, we recall some results related to Finsler manifold. Let X be a real Banach Space. /, gl, g2 . .. gn : X --+ Rl be in C l ,

M = {x EX : gi(X) = O,i = 1,2", ·,n}, where {g~(x)}r are linearly independent for any x E M, then M is a submanifold of X with the natural Finsler structure (see [26]), hence it is a complete Finsler Manifold. Also, for any p E M there is a coordinate neighborhood in M which is diffeomorphic to {g~(p),gHp)"" ,g~(p)}.J.., i.e. there is the direct sum decomposition X = Xl ffispan{el,e2,· ·· ,e n } such that there is a neighborhood U of p in M such that U is diffeomorphic to a neighborhood V of () EX, where ei E X satisfies

and the tangent space Tp(M) of M at p satisfies Tp(M) we have the following lemmas: Lemma 2.5. (Lemma 2.4 of [16]) have

~

Xl' FUrthermore,

With the same notations as above, we

n

(d/IM(p),7rV) = (f'(p),v) -

I)!' (p),ej)(gj(p),v)

for all v E X,

j=l

where

11'

is the projection of x to Tp(M)

~

Xl.

The following result is the Ekeland's variational principle on Finsler manifold.

Positive solution to p-Laplacian type scalar field

197

Lemma 2.6. (Corollary 1.3 in Chapter 2 of [26]) Let M f:. be a complete Finsler manifold, f E Cl(M,R) be bounded from below, then there is a sequence {Pn} C M such that

In particular, if f satisfies (PS) condition, then there is a Po E M such that f(Po)

= pEM inf f(p)

and df(po)

= 0.#

Now we study A defined by (1.7). By f(t) E C 1 (R), for any u E W 1 ,P(R N ) the functional defined by

is a C 1 functional and for any u E A, we have u+ for all s < 0 and u -:t. o. Hence

(g'(u),u)

r (lV'uI iRN

-:t. 0 by the facts of f(x, s) == 0

r [f(u)u + f'(u)u iRN

=

p

=

r [(p iRN

l)f(u)u - f'(u)u 2 Jdx

r [(p iRN

l)f(u+) - f'(u+)u+Ju+dx < 0,

P

+ mlulP)dx -

2

Jdx

by (C6).

So A is a closed hence complete submanifold of W 1 ,P(R N ) with the natural Finsler structure. Therefore, using Lemma 2.5 with el = (g'(~)'u) we have Lemma 2.7.

For any u E A, Vv E W 1 ,P(RN

(dJ OO IA(U),1TV)

),

(loo' (u),v) - (loo' (u),ed(g'(u),v)

=

(loo' (u), v)

(by

g(u)

= 0,

Vu E A),

where 1T is the projection from W 1 ,P(RN ) to TuA.# By Lemma 2.6, Lemma 2.7 and noting A f:. 0 (see Lemma 3.1 in Section 3), we have Lemma 2.8. There is a sequence {Un} C A such that

JOO(u n ) ~ Joo

= uEA inf JOO(u)

and

J oo ' (un) ~ 0 in W- 1 ,p' (R N ).#

198

Gongbao Li, Lina Wu and Huan-Song Zhou

3

Proof of Theorem 1.1 Before proving Theorem 1.1, we give two lemmas. Lemma 3.1. If (C1)-(C3) (C5) hold and l > m, then A -::f. Proof. We denote

For a

cp.

N

> 0, setting wa(x) = (d(N))-laP'I e-alxIP, it is easy to get that Iwa(x)lp

= 1,

l\7wa(x)l~

= aD(N)

and

then by (2.2), it is easy to see that

(I' (twa (x)), twa(x)) 2::

° for

On the other hand, if we choose a E (0,

t>

° small enough.

(3.1)

b(J:;)), where l is given by (C3), then (3.2)

and by Fatou's lemma

IlwaliP-

lim

{

t--t+oo ) RN

!(x, twa)twa dx tP

lim !(x, tWa~W~ dx tP-Iw~ 1

< IlwallP - {

) RN t--t+oo

= =

Ilwall P -llwal~ l\7wal~

+ (m -l)lwal~ < 0,

by(3.2).

So (I'(tw a ), tWa) -+ -00 as t -+ +00, this and (3.1) imply that there is at· > large enough such that (I' (t·w a ), t·w a ) = 0, t·w a E A.#

°

Proposition 3.1. (Theorem 5 in [22]) Let 0 be a smooth domain in RN (possibly unbounded) , u E C 1 (0) be such that ~pu E L~oc(O), u 2:: a.e. in O. ~pu S f3(u) a .e. in 0 with f3 : [0, +00] -+ R continuous, nondecreasing, f3(0) = and either f3(s) = for some s > or f3(s) > for all s > but 1 ' Jo1(j(s))-pds = 00 holds, where j(s) = J; f3(t)dt. Then if u does not vanish

°

°

°

°

°

°

199

Positive solution to p-Laplacian type scalar field

identically on fl, it is positive everywhere in fl .# Lemma 3.2. If (C1)-(C3) and (C6) hold, then Joo > O. Proof. If Joo = 0, by Lemma 2.8 there is a sequence {un} C A such that

(3.3)

First of all, we show that {un} is bounded in Wl'P(RN). By contradiction, we suppose that (3.6) lIu n ll -+ +00 as n -+ +00, and for any fixed a > 0, let

Clearly, {w n }, {w;i} are bounded in Wl,P(R N ). For Pn(x) = Iw;t(x)jP, the conditions of Lemma 2.2 are satisfied. We claim that neither (i) nor (ii) of Lemma 2.2 holds. Hence, (3.6) can not be true, that is, {Un} is bounded in W1 ,P(R N ). In fact, if there is a subsequence of {Pn}, still denoted by {Pn}, such that Lemma 2.2 holds. We get contradictions in both cases. Case 1

Vanishing: In this case, by Lemma 2.3 and (2.3), we see that

So

1 1 P P = -lIw n ll + 0(1) = -a + 0(1), p p

by(3.7)

(3.8)

where, and in what follows, we denote by 0(1) the quantity which tends to 0 as n -+ +00. But, by Lemma 2.4 and (3.3)

(3.9)

Gongbao Li, Lina Wu and Huan-Song Zhou

200

which is impossible by (3.8) . Case 2 Nonvanishing: In this case, there are 1] > 0, R RN such that lim ( Iw;t(x)IPdx 2: 1] > O.

> 0 and {Yn}

C

(3.10)

n--++oo lYn+BR

Set w;t(x) = w;t(x + Yn), then IIw;t(x)11 = Ilw;t(x)11 ~ IIwn(x)11 = a and by Sobolev imbedding we may assume that for some w(x) E Wl,P(R N )

w;t(x) ~w(x) weakly in W1,P(R N ), w;t(x) ~w(x) strongly in Lioc(R N ), w;t(x) ~w(x) a.e. in RN , these and (3.10) imply that

w(x)

't 0

and w(x) 2: 0 a.e. in RN.

(3.11)

By (3.6) (3.7) and (2.6), for n large enough, we have tn =

a

Ilunll

() l(tnu;t) !Cu;t) E 0,1 and (tnun)p-l + < + ' - (Un)p-l

hence

then it follows from (2.6) and Fatou's Lemma that

= p~lIwnliP >

( P(wn)dx 2: l [~l(w;t)w;t hN hN P

inN [~!Cw)w - pew)] dx + 0(1).

So, by (2.6) and (3.9) we have

P(w;t)] dx

Positive solution to p-Laplacian type scalar field

201

this means tV == 0 which contradicts (3.11). Thus, {un} is bounded in W1,P(R N ). Letting Pn(x) = lun(x)IP and by Sobolev imbedding that {Pn} is bounded in Ll(RN), then applying again Lemma 2.2 we know that for some subsequence of {Pn(x)} either Vanishing or Nonvanishing occurs.

Case I

If Vanishing occurs: Similar to (3.8), we have

1 [OO(u n ) = -lIunll P + 0(1).

(3.12)

P

Taking

€

= T in (2.2), it follows from Ilunll P = £N

this means that

(3.4) that

f(un)undx ::::;

1 211unilP : : ; Cllunll q

~llunllP + Cllunll q ,

(q > p). Hence, there is a fJ > 0, such that (3.13)

[00 (Un) ~ Joo

So, if

Case II such that

= 0, (3.12) and (3.13) are contradictory.

> 0, R > 0, {Yn}

If Nonvanishing occurs. There exist TJ lim n-too

( lun(x)IPdx iyn+BR

~ TJ > O.

C RN

(3.14)

Let un(x) ~Un(X + Yn}, we claim that for any cP E COO(RN),

{

iRN Indeed, so

[lV'u n IP- 2V'u n V'cp + mlun lp - 2u ncp] dx - (

Vcp E Wl'P(RN), let CPn = cp(x -

1([00 (Un), cp)1 1

f(un)cpdx

iRN

Yn}, it is easy to see that

=

I£N [lV'unIP- V'unV'cp +

=

1£)V'unIP-2V'un V'cpn

2

p

mlunl

-

(3.15)

IICPnl1 = Ilcpll,

2 uncp - f(un)cp)dxl

+ mlunlp - 2unCPn - f(un)CPn)dxl

1 1(I001 (un), CPn)1 : : ; 11[00 (un)IIIICPnll = 11[001(un)llllcpll ~ 0 by (3.5), and (3.15) is proved.

= 0(1).

Gongbao Ll~ Lina Wu and Huan-Song Zhou

202

Since {un} is bounded in W1,P(R N ), {Un} is also bounded in W1,P(R N ), then we may assume, by Sobolev imbedding, that for some u(x) E Wl,P(R N ),

~ u(x) weakly in W1,P(R N ), Un (x) ...2:.t u( x) strongly in Lioe (RN), { un(x) ...2:.t u(x) a.e. in R N , un(x)

(3.16)

and u(x) t. 0 by (3.14). By (3 .16) and (2.2), it is clear that for any t.p E W1,P(R N ),

(

iRN

[/(Un) - /(u)]t.p(x)dx...2:.t 0 and (

iRN

m[lu nlP- 2u n - luIP-2U]t.pdx...2:.t o.

On the other hand, by (3.16) and (3.5), we can show that for any t.p E W1,P(R N ), { [lVu IP - 2 Vu - IVuIP- 2Vu]Vt.pdx...2:.t 0,

iRN

n

n

which is trivial if p = 2, but for general p compactness principle to show first that

> 1 we need to use the concentration(3.17)

(see the proof of Theorem 1.6 in [23] for details) and then the result follows. So U is a weak solution of -Llpu + mlulp-2u = /(u), that is,

{

iRN

[lVuI P- 2VuVt.p+mlul p- 2ut.p]dx- {

iRN

/(u)t.pdx = 0, for any t.p E Wl'P(RN). (3.17')

Taking t.p = u - in the above formula, it is easy to see that u - == 0 by noticing (Cl), hence u ~ 0 a.e. in RN and by standard regularity results [29] u E CI~~(RN) . Using Proposition 3.1 with {3(u) = mlul p - 1 , and noticing Llp(u) = mlul p- 2u - /(u) ~ {3(u), we see that u is positive everywhere in RN. Then using again (2.6) and Fatou's lemma, we have

which means that Nonvanishing is also impossible. So, Joo > 0.# Proof of Theorem 1.1. By Lemma 3.2, to prove Theorem 1.1, it is enough to show that Joo is achieved by some Uo E W1,P(R N ) and Uo > 0 a.e. in RN.

Positive solution to p-Laplacia.n type scalar field

203

Just like in the proof of Lemma 3.2, it follows from Lemma 2.8 that there exists {un} C W1 ,P(R N ) such that (3.3)-(3.5) hold, and keep in mind that Joo > 0 from now on. Step 1 {un} is bounded in Wl ,P(R N ). If

lIunll ~ +00, we let

Set Pn(x) = Iwn(x)IP, ifthere is a subsequence, still denoted by Pn(x), such that Lemma 2.2 holds, then by the same processes as in getting (3.8) (3.9), and noticing that (3.19) we know that Vanishing doesn't occur. If Nonvanishing occurs, i.e. there are {Yn} C RN, R

Wn(X) ~ wn(x

weakly in Wl,P(RN),

wn(x) ~ w(x)

strongly in Lroc(RN) p::; q

wn(x) ~ w(x)

a.e. in RN

= un(x + Yn),

> 0 such that

=!. 0 satisfying

Wn(x) ~w(x)

Set un(x) we get:

'1/

+ Yn), Ilwn(x)11 = Ilwnll = p(Joo ) t,

hence , there is some 'Ill E W1,P(R N ) and 'Ill

{

> 0,

then wn(x)

= ilfi~(m,

< p.

= :!!p,

(3.20)

and 'VIP E C(f(RN) by (3.15)

that is, 'VIP E C(f(RN),

r [I'VwnIP-2'Vwn'VIP + mlwnl iRN

p

where

Pn ( X ) =

-

2w IP]dx n

llid _p_l , Un

{ 0,

r

iRN

Pn(x)(w;t)P-1IPdx

= 0(1), (3.21)

204

Gongbao Li, Lina Wu and Huan-Song Zhou

Similar to (3.17) we have \7w n ~ \7w a.e. in RN, then Vtp E Co (RN) by (3.20) (3.21)

{ [I\7wI iRN

P - 2 \7w\7tp+

iRN Pn(x)(w~)P-ltpdx+o(l).

mlwl p- 2wtp)dx = (

(3.22)

By (2.5) (2.6), 0 :::; Pn(x) :::; £, {Pn(X)} C LOO(RN) and there is a hex) E LOO(RN) with 0 :::; hex) :::; £ a.e. in RN such that Pn(x).3. hex) weakly· in LOO(R N ). Since wn ~ w strongly in Lroc(RN), we get ...l!...-

Pn(X)(W~)p-l .3. h(x)(w+)p-l weakly in Ll~~' (RN),

{

iRN

Pn(x)(W~)P-ltpdx ~

(

iRN

h(x)(W+)p-ltpdx, Vtp E CO(R N ).

Thus by (3.22) and the density of Co(RN) in W1 ,P(R N ) we have for all tp E W1 ,P(R N ) that

{

iRN

[I\7uijP-2\7w\7tp

+ mlwl p- 2wtp)dx =

(h(x)(W+)P-ltpdx,

iRN

(3 .25)

taking tp = w- in the above formula, it is easy to see that w- == O. So, w a.e. in RN and w is a weak solution of

~

0

Applying Proposition 3.1 as in the end of the proof of Lemma 3.2, we see ~

> 0 a.e. in RN. Noticing wn(x) = p(I~:iI Un ~ w(x) > 0 a.e in RN, then un(x) ~ +00 a.e. in RN and Pn(x) = ~~"::( ~ £, that is, hex) == £ Un

that w(x)

a.e. in RN and (3.25) implies that

{

iRN

l\7wl p- 2\7w\7tpdx

= (£ -

m) (

iRN

IwIP-1tpdx,

which is impossible since there is no nontrivial solution to -~pu = in W1,P(R N ) for any>. E Rl by Lemma 2.1, and Step 1 is proved.

>'lulp-2 U

Step 2 Joo is achieved by some Uo E Wl,P(R N ) and Uo > O. Let Pn(x) = lun(x)IP, by extracting a subsequence, we may assume that {Pn} satisfies Lemma 2.2. If Vanishing occurs, then by using Lemma 2.3 and (2.2),(2.3), we have

Positive solution to p-Laplacian type scalar field

205

Again by (3.4), we get

[OO(u n )

=

IN [tf(Un)Un - F(un)1dx ~

0,

which contradicts (3.3) since we have now Joo > O. So only Nonvanishing occurs. Then following the same procedures as in (3.14)-(3.17)1, there exists u(x) E WI ,P(R N ) with u(x) t= 0 such that

r [lVu(x)I JRN

P 2 - VuV p.

c = inf max [(-y(t)), -yEro p such that l(ud < O. Proof. Taking c = T in (2.3) and by using Sobolev imbedding, it is easy to see that the first part of this lemma is true. Now, let uo(x) > 0 be the solution of problem (1.5) obtained by Theorem 1.1, then loo(uo) = Joo > O. (4.1) For any fixed a

> 0, setting ua(x)

X = uo(-), a

(4.2)

then

l\7ual~

= aN-"I\7uol~,

IUal~

= aNluol~·

(4.3)

So by the definition of 100 given by (1.6), we have

(4.4)

Since N > p, we can find ao > 0 large enough such that

Hence, J(u ao ) < loo(u ao ) < 0 by (C5), and Lemma 4.1 is proved for Ul Lemma 4.2.

= u ao '#

Under the same conditions and u ao as Lemma 4.1, define c = inf max l(')'(t)) , "(HtE[O,l]

(4.5)

Positive solution to p-Laplacian type scalar field

207

where r = bE C([O, 1], Wl,P(R N )) : ')'(0) = 0, ')'(1) = uo: o }, then c E (0, JOO) . Proof. By Lemma 4.1, c ~ f3 > O. Let ')'(t) = uO(o:~t) for t E [0,1], it is easy to see that 1I')'(t)1I -+ 0 as t -+ 0, then we may set ')'(0) = 0 and ')'(t) is a continuous curve joining 0 and uo: o ' By (4.5) and the continuity of I(u), there exists E (0,1] such that

t*

=

c ~ sup I(')'(t)) tE[O,l]

=J

(UO(o::t*))

x sup I(uo(-)) O:ot

tE[O,l]

0, R > 0 such that lim n-too

Let wn(x) = wn(x + Yn), with W t; 0 such that

r

lYn+BR

Iwn(x)IPdx

Ilwn(x)11 = Ilwnll,

~ TJ > O.

then for some w(x) E Wl,P(R N)

(4.12) (4.13) Wn(x)

....!:.t w(x)

a.e. in RN.

(4.14)

Let un(x) = un(x + Yn) and for any cP E W1,P(R N ) setting CPn(x) then IICPnl1 = Ilcpll, similar to (3.15) we have

I(I'(un),CPn)1

= cp(x -

< III'(un)llw-l ,P'(RN)IICPnll

=

III'(un)llw-l.P'(RN)llcpll -t 0

hence

that is, for any cP E W1 ,P(R N

)

we have

as

n -t +00,

Yn),

Positive solution to p-Laplacian type scalar field

_j

RN

I(x + Yn, un) ()P-I d -p-l Wn cp X Un

209

0

(1)

i.e.

where I(x Pn (X ) =

{

+ Yn, un) -p-l' Un

0,

if Un> 0, if Un:::; O.

By (C3) (C4), we know that 0 :::; Pn{x) :::; l, then, by extracting subsequence, there exists some hex) : 0:::; hex) :::; l such that Pn(x).2l hex) weakly· in Loo.(O). Noticing . LPIDe (RN) ' strongIy In .J!...-p-l n -p-l strongly in LI~~l (R N ), wn -+ W -p-l n -p-l a.e. in RN . wn -+ W

-+

n Wn W-

Hence Pn{x)( w;t)p-l .2l h{x) (W+)p-l in L;I!::r (RN) and similar to (3.17), V'wn ~ V'w a.e. in R N , then

Thus

r [lV'wI P- 2V'wV'cp+mlwl p- 2wcpjdx r h(x)(W+)V-Icpdx, 'Vcp =

iRN

E WI,V(R

N ),

iRN

taking cp = w- , it is easy to see that w- == O. So W ~ 0 a.e in R N , that is, w is a weak solution of -div(lV'ulp- 2 V'u) + mlulp-2 u = h(x)(u+)P-l, by Proposition

210

Gongbao Li, Lina Wu and Huan-Song Zhou

3.1 and 6 p (u) a.e. in RN. Since wn N R , hence

= mlulp-2u 1.

= PIILi!

Pn () X

h(x)(U+)P-l ~ f3(u) ~mluIP-l, we see that W > 0

~ W > 0 a.e. in R N , we have that Un ~

Yn, un) = f(x +Un -p-l

+00

a.e in

n n n --+ (., that is, h(x) == (. a.e. in

So we get

which is impossible since there is no nontrivial solution for -6 p u = Alulp-2u for any A E Rl by (2.7), then {un} is bounded in Wl,P(R N ) and it is easy to see that Step 1 is proved by extracting a subsequence of {un} and some Uo E W1,P(R N ).

By Step 1, (2.2) (2.3) and the main results of [23] we have that V'U n ~ V'uo a.e. in RN and I'(uo)

= o.

It remains to show that Uo E Wl,P(R N) \ {O} . By contradiction, assume that Uo == O. If Un ~ 0 strongly in Wl ,P(RN), then lim [(un) = 0 which contradicts

lim [(un) = C > O. So Un n ..... oo that for some TJ > 0

n ..... oo

f+ 0 in Wl,P(RN) as

n ~

00

and we may assume

To finish the proof of Step 2, we need the following lemma: Lemma 4.3. Let (Cl)-(C6) hold, if {Uk} such that Uk ..!s. 0 in Wl ,P(R N ), then

Proof of Lemma 4.3.

c

Wl ,P(R N ) is a sequence

For any given R > 0 we see that

211

Positive solution to p- Laplacian type scalar field

~ I~ + I~ Since Uk .!:.. 0 weakly in W1,P(R N ) , by Sobolev's imbedding we know that Uk ~ 0 strongly in Lfoc(RN) for some subsequence of {Uk} (still denoted by {ud) and q is given as in Step 1. Hence (2.3) implies that lim I~ =

k-too

o.

On the other hand, for any given 8 > 0,

I~

[!

=

Izl>R

{lu~IR {6:5IUkI9-'}

+!

I-I>R

{lu~li6-1}

jlF(X' Uk) -

P(uk)ldx

< cl(8) {

( IUkIPdx+C3(8) ( IUklpodx iRN IUkIPdx+C2(R)8-P iRN iRN

where cl(8)

=

IF(x'~~I~ F(t)1

sup

- t 0 as 8 -+ 0+, by (C2) ,

{~~I~:5; }

c2(R)

=

IF(x, t) - P(t)1 - t 0 as R - t +00 for fixed 8, by (C5),

sup I·I>R } { 6:5':5-6- 1

C3(8)

=

IF(x,

sup {

1-lo::R }

'0:: 6 -

1

Therefore, lim I~ k-too

?I; P(t)1 - t 0 as 8 -+ 0+ , p. = ;~ , by (C3). t p

= 0 and hence

Similarly, Lemma 4.3 is proved. Now we turn to the proof of Step 2. By Lemma 4.3 we have

c = I(Uk)

+ o~) = IOO(Uk) + 0(1)

{ (Ioo'(Uk),Uk) =J.Lk

= 0(1).

(4.16)

Gongbao Li, Lina Wu and Huan-Song Zhou

212

Let Uk(X)

= Uk(tkX),

where tk > 0 will be determined later, then (4.17)

(1 00 ' (Uk), Uk)

= t;;N (t~ - l)JRN IV'UklPdx = t;;N[(t~ - 1) JRN

+ t;;N (JOO' (Uk), Uk) IV'UklPdx + ILk] .

(4.18)

We claim that there is a A > 0 such that (4.19)

In fact, if (4.19) is false, then this implies that

JrRN IV'UklPdx ~ 0,

r

mluklPdx

Noticing !(X,Uk)Uk :S ~IUkIP

+ CIUkIP·,

JRN

but lim k--+oo

IIUkll P = rl',

~ fJP > O. we have that

this contradiction shows that (4.19) is true. Therefore we can set tk

=

(1 + JRN iV':k

1

IPdx ) P

such that Uk E A and

tk ~ 1. This facts together with (4.16) (4.17) imply that

which contradicts the fact that c < Joo (see Lemma 4.2). We have thus proved that Uo E WI ,P(R N ) \ {O} . Finally, by the same procedures as in the end of the proof of Lemma 3.2, we see that Uo > 0 in RN by Proposition 3.1. The proof of Theorem 1.1 is complete.#

Remark 4.1. We mention that under the assumptions (Cl)-(C6) we could actually show that Uk ~ Uo in WI ,P(R N ) and l(uo) = c by using the concentration-compactness principle as in [16] .

Positive solution to p-Laplacian type scalar field

213

References

1 A. Ambrossetti and R. Rabinowitz, Dual variational method in critical point theory and applications, J. Funct. Anal, 14(1973), 349-38l. 2 H. Berestycki and P.L. Lions, Nonlinear Scalar Field Equations, I: Existence of a Ground State, Arch. Rat. Mech.Anal., 82(1983), 316-338. 3 H. Berestycki and P.L. Lions, Nonlinear Scaler Field Equations, II: Existence of Infinitely Many Solutions, Arch. Rat. Mech . Anal., 82(1983), 347-369. 4 W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. in Math. Phys., 55(1977), 149-162. 5 N. Berger and M. Schechter, Embedding theorems and quasilinear elliptic boundary value problems for unbounded domains, Trans. Amer. Math. Soc., 172(1972), 261-278. 6 P.L.Lions Principale de concentration-compacite en calcul des variations, C. R. Acad. Sci. Paris, 294(1982), 261-264. 7 P.L. Lions, La methods de concentration-compacite en calcul des variations. In Seminaire Goulaouio-Neter-Sohwarlz, 1982-1983, Ecole Polytechnique, Palaiseau, 1983. 8 P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1, Ann. 1. H. P. Anal. Nonli., 1(1984),109-145. 9 P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 2, Ann. 1. H. P. Anal. Nonli., 1(1984)4, 223-283. 10 P.L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1, Revista Matematica Iberoamericana, 1(1985)2, 45-120. 11 C.A. Stuart, Bifurcation for Dirichlet Problems without eigenvalues, Pmc. London Math. Soc., 45(1982), 149-162. 12 J.F. Yang and X.P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains, (I) positive mass case, Acta Math. Sci., 7(1987), 341-359; (II) The zero mass case, ibid, 8(1988),447-459. 13 P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. in Math., No. 65, AMS, Providence, R.1.l986. 14 C.A. Stuart, Self-trapping of an electromagnetic field and bifurcation from the essential spectrum, Arch. Rat. Mech. Anal., 113(1991), 65-96.

214

Gongbao Li, Lina Wu and Huan-Song Zhou

15 C.A. Stuart, Magnetic field wave equations for TM-modes in nonlinear optical waveguides, Reaction Diffusion Systems, Edit, Caristi and Mitidieri, Marcel Dekker, 1997. 16 G.B. Li and H.S. Zhou, The Existence of a Positive solution to asymptotically linear scalar field equations, Pmc. Royal Soc. Edinburgh, Section A, 130A(2000). 17 D.M. Cao, G.B. Li and H.S. Zhou, The Existence of two solutions to quasilinear elliptic equations on R N , Chinese J. Contemporary Math., 17(1966), 277-285. 18 L. Jeanjean, On the existence of boundary Palais-Smale sequences and application to a Landesmann-Lazer type problem, Pmc. Royal Soc. Edin., Section A, 129A(1999), 787-809. 19 D.G. Costa and O.H. Miyagaki, Nontrivial Solutions for Perturbations of the p-Laplacian on unbounded domains, J. Math. Anal. Applications, 193(1995),737-755. 20 C.A. Stuart and H.S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on R N , Comm. in P. D. E., 24(1999), 1371-1785. 21 H.S. Zhou, Positive solution for a semilinear elliptic equation which is almost linear at infinity, Z. angew. Math. Phys, 49(1998), 896-906. 22 J.L. Va'zquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12(1984), 191-202. 23 G.B. Li, The existence of a weak solution of quasilinear elliptic equation with critical sobolev exponent on unbounded domains, Acta. Math. Sci., 14(1994)1,64-74. 24 H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence offunctionals, Proc. Amer. Math. Soc., 88(1983),486-490. 25 M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. TMA, 13(1989), 879-902. 26 K.C. Chang, Critical Point Theory and Its Applications, Shanghai Scientific and Technical Press, 1986, (in Chinese) . 27 G.B. Li and H.S. Zhou, Asymptotically "linear" Dirichlet problem for pLaplacian, to appear in Nonli. Anal. 28 C.A. Stuart and H.S. Zhou, A variational problem related to self-focusing of an electro-magnetic field, Math. Meth. Appl. Sci., 19(1996), 1397-1407. 29 P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eq., 51(1984), 126-150.

The existence and convergence of heat flows * Jiayu Li t

1

Introd uction

Let M and N be two compact Riemannian manifolds with dim M = m and dim N = n. In this paper we consider the existence and convergence of heat flows for harmonic maps. It was shown in [8] that the heat equation has a unique local solution, and that if the sectional curvature of N is nonpositive, it has a global solution and the solution converges to a harmonic map as t -+ 00. However (see [5], [7], [3], and [2]), in general the solution may blow up at a finite time or at infinity. Recently Lin-Wang [11] considered the exis~ence and the regularity property for the weak heat flow. They proved that if N does not carry a harmonic sphere S2, for any Uo E Hl,2(M, N), there is a stationary weak heat flow with initial data uo. We will give a new proof here which depends on the ideas developed in [13] . Furthermore, we show that the stationary weak heat flow subconverges to a stationary harmonic map. The main result can be stated in the following theorem.

Theorem 1.1 If the target manifold N does not carry any harmonic spheres S2, for any initial data Uo E Hl,2(M, N), there is a stationary weak heat flow u(x, t) with initial data uo, and u(x, t) subconverges to a stationary harmonic map u(x) as t -+ 00 . "This work is supported by NSF grant . tlnstitute of Mathematics, Academia Sinica, Beijing 100080, P. R. China.

215

Jiayu Li

216

For simplicity in notation, we assume in this paper that the metric on M is flat, i.e. we assume that M = Rm.

2

Strong convergence of the stationary weak heat flows

We recall the definition of a stationary weak heat flow, the two monotonicity inequalities (see [13]), and The to-regularity.

Definition 2.1 We say that u(x, t) E H 1,2(Rm x R+, N) is a stationary weak heat flow, if it is a weak heat flow and for almost all t > 0, (1) for any smooth vector field X on Rm with compact support,

(2) for any

dt



E Co(Rm) with

j . 2 -

i

to - r21 1

to-r~

Rm (to

aU

'Vu (Y - X ) 2 2(to _ t) ) G(x,to)(Y, t)dydt.

- t)( at -

We set

E(u, (x, t), r) = r 2- m

( Br(x)

J

I 'V u(y, t)1 2 dy

and

J(u, (x, t), r) = T2- m (

JBr(X)

lat ul 2dy.

Lemma 2.3 Suppose that J(u, (x, t), r) :::; T- 2a for 0 < r < TO, where 0 < a < 1. Then faT any 0 < rl < r2 < TO,

The Eo-regularity for smooth heat flows was proved by Struwe [15]. The partial regularity for stationary weak heat flows into spheres has been studied by Chen-Li-Lin [6] and Feldman [9], for the flows into a general target, the same result was proved by Liu [12] using the ideas in [6] and [1]. The theorem is as follows:

Theorem 2.4 Let u(x, t) be a stationary weak heat flow. Set

I 1 t r2

1 P(u, (x, t), r) = ~ r

There exists

EO

+

t-r2

Br(x)

> 0, such that if P(u, (x, t), r)
0 depends on m, N, EO, E(u(·, 0)). Suppose that Uk is a sequence of stationary weak heat flows for harmonic maps with

By (2) in Definition 2.1, we have

(3) So we may assume that Uk --t U weakly in H I ,2(Rm x [0,00), N) . We set Et = nr>o{x E RmlliminfP(uk, (x , t),r) k--too

2:

EO}

to be the blow up set for the sequence Uk at t, and set E = Uor>O {x

So(t)

n

for any

E Rmllim inf I( Uk, k-HlO

(x, t), r) 2: r- 20 },

> 0,

€

Lemma 2.5 The measure of A , IAI is O. For any t E (0, (0) , we have Hm-2('L}) < +oo.Ift f/. A, and 0 < a < 1, Hm-2(1+o)(So(t)) < +00, and Hm-2(S;(t)) = 0 for any f > O. In this section, we mainly prove the following theorem which was first proved by Lin-Wang ([11]). We will use the ideas developed in [13] to give a new proof.

Theorem 2.6 Let Uk be a sequence of stationary weak heat flows for harmonic maps from Rm to N with energies E(Uk(', 0)) ::; A. If the target manifold N does not carry any harmonic spheres S2, there is a subsequence of Uk which we also denote by Uk such that Uk --+ U in Hl~;(Rm x R+, N) . And U is a stationary weak heat flow. Proof: It is clear th,at we may assume that Uk --+ U weakly in Hl ,2(Rm x R+, N) . The following lemma is a monotonicity inequality for the measure l/t ([13]). Lemma 2.7 Ifliminfk-+oeJ(ukl(x,t),r) ::; r- 2o , for 0 < r < ro, then for any 0 < rl < r2 < ro, we have

r~-m

r JB

I \l uI 2dV + r~-ml/t(Br2(X))

T2 (X)

~ r~-m

r

} BTl

I \l ul 2dV + r~-ml/t(Brl (x)) - c(r~-O - d- o ), (x)

where c > 0 depends on m, A and a

Jiayu Li

220

We proved the rectifiability for Et in [13]. Theorem 2.8 Let Uk be a sequence of stationary weak heat flows for harmonic maps from Rm to N with energies E(Uk(', 0)) ~ A. Let Et be the blow up set at t > O. Then it is a closed set. If t ~ A, Et is a rectifiable set. It is not difficult to see by Lemma 2.7 that, if t ~ A, Vt = O(x, t)Hm- 2l Et) where O(x, t) is upper semi-continuous in Et \ ScAt) with C(EO) ~ O(x, t) ~ C(A) for Hm-2_a.e. x E E t , C(EO) is a positive constant depending only on m and EO, C(A) is a positive constant depending only on m and A. So, O(x, t) is Hm-2 approximate continuous for Hm-2_a.e. x E Et. In the sequel, we assume that Uk does not converge strongly in Hl~; (Rm x R+) N), in other words, there exists t > 0 such that Vt(E t ) > O. We will show that we may derive a harmonic sphere S2 using the blow up analysis. We set B = {t E R+IVt(Et) > O} . By Corollary 6.2 in [13] we have B = (0, T) where T < 00, or T = 00 . The following Geometric Lemma was proved in [13]. Lemma 2.9 Suppose that x E Et \ Sn(t) and O(x, t) is Hm-2 approximate continuous at x. Then there exists r x > 0, such that, for any 0 < r < r x , we can find m - 2 points Xl, ... , Xm-2 in Br(x) n (Et\Sn(t)) so that, (1) IO(xj, t) -O(x, t)1 < Er , for j = 1"", m-2, where Er -+ 0 as r -+ 0; and (2) IXII 2': Smr, dist(xk, X + Vk-d ~ Sm r , for k = 1" . " m - 2, where Vk- l is the linear space spanned by {Xl - X,"', Xk-l - x}. Here Sm > 0 depends only on m . Using the Geometric lemma, we show the following lemma in [13] . Lemma 2.10 Let T E TEt . 1ft fj. A, we have lim lim f-+O k-+oo

JrB.(Et) I \iT ukl2dy =

O.

Here and in the sequel we denote by Bf(X) = {x E Rm I dist(x, X) < E}.

The existence and convergence of heat flows

221

Set Fk€(x)

=

r I VT ukI 2(x, x')dx' JB~(O)

for x E Et. Here and in the sequel, we denote by B;(x) the metric ball centered at x with radius r in R2. We consider the HardyLittlewood maximal function MFkf (x) of FkE{X)' which is defined by MFkf{X)

=

sup r 2 -

r

m

O 0, H

m

-

2 (U:=1 n~=no Urc;=l n~ko {x

E EtIMFk (!,) (x)

1

~ l}) = o.

By the partial regularity result (cf. [6], [9], [12]), we can find t E R+ and Xl E Et \ (Sdt) U S;(t)) C B~-2(O), such that for any no > 0 and any ko > 0 there are nl > no and kl > ko satisfying 1

MFk,( "i) 1 (Xl) < -l.

(4)

and Uk(X, t) is smooth near (Xl, X') for all x' E B?(O), and (Xl, X')

rt

Sdt). 2 We claim that , for all k sufficiently large, we may find Ok --+ 0 such that max Ok2-m x/EBf(O)

1

B;:'-2(Xk)XB~k (x')

I V Uk 12( X, X')d XdX

I

= 8. to22m .

(5)

Jiayu Li

222

In fact, since Uk(X, t) is smooth at (Xk' x'), for any given k and for 0 < o(k), we have

If x E Et \ S;(t), we have

lim E( Uk, (x, t), r) ;:::

k-too

1 r -2 lim inf P( Uk, (x, t), -) 4 . 2m k-too 2

C€ -

c-/i.

So, for fixed 0 > 0 and sufficiently large k,

t

o2-m

} B;-2(Xk) x Bg(0)

I \l ukI2(X, x')dxdx';:::

to . 4· 22m

Therefore we can choose Ok > 0 so that (5) holds. By (4) and (5), since (Xk' x') f{. S!(t), we can find Ek --+ 0, rk --+ 0, 2 (Xk' xD E Et and a subsequence of Uk, which we also denote by Uk for simplicity, such that rk- m )

rB~-2(Xk) X B;k(xU I \l ukI 2(x, x')dxdx' = 8· to22m

The existence and convergence of heat flows

MFk 2r 2 - m r Iau 12 da _ 2r 1 - m r (r au(y , t k)) au(y, tk) dy . J 8B r ar J Br{x) ar at

The existence and convergence of heat flows

227

By Holder inequality, we get

dE(u, (x, t k), r) dr 1 > 2r 2ml18U2 -8 I dcr-2(E(u,(x,t k),r))2(I(u, (x,tk),r))2.1

r

{)B r

Integrating with respect to r from k --+ 00, we get

r~-m

( } B r2 {x)

~ r~-m

rl

to r2 (0
O. By Lemma 3.4, we have

thus for any l > 0,

Hm-2(U~=1 n~=no Uko=l n~ko {x

E EIMFk(;t) (x)

2':

~}) = o.

By the partial regularity result (cf. [6], [9], [12]), we can find Xl E E \ 512 (tk) c B~-2(0), such that for any no > 0 and any ko > 0 there are nl > no and kl > ko satisfying

(13) and u(x, t k) is smooth near (Xl, x f) for all Xf E Br(O), and (Xl, x f ) fI51 (tk). 2 We claim that, for all k sufficiently large, we may find 8k -t 0 such that max

X'EB~(O)

8~-m

r

} B";',.-2(Xk) XB

'ik (x')

I \l uI 2(x, Xf, tk)dxdx f =

8 .

E~2

.

(14)

m

In fact, since u(x, tk) is smooth at (Xk' Xf), for any given k and for 8 < 8(k), we have

On the other hand, by (12) we have EO

lim E(u, (x, tk), r) 2': 4 . 2m -2·

k-+oo

230

Jiayu Li

So, for fixed

~

> 0 and sufficiently large k,

Therefore we can choose ~k > 0 so that (13) holds. By (13) and (14), since (Xk' x') tt. Sl2 (t k), we can find Ek --t 0, Tk --t 0, (Xk, X~) E E and a subsequence of Uk , which we also denote by Uk for simplicity, such that

MFk 2, M > 0 such that

o < (}G(x, t)

~

t . g(x, t)

for

It I ~

M.

(g3) is a famous superlinear condition given by P. Rabinowitz. The first important result for (1.1) is the following. Theorem 1.1 Let (gt) - (g3) hold. The (1.1) has at least one nontrivial solution. This theorem was given by A. Ambrosetti and P. Rabinowitz (see [AmR1]). In [Cha2], K. C. Chang weakened the conditions (g2), (g3) by -1' g(i' t) < Imt-+O _

\

t

uniformly in xED,

+t

uniformly in xED,

Al -

· -+ g(i' t) > \ 1Im _ Al t o

and

(}G(x, t)

~

tg(x, t),

"Ix E 0 as

It I ~

M.

More complicated problem is the following - 6. u { u = 0,

+ a(x)u = g(x, u),

in 0 on aD

(1.2)

where a(x) E LOO(O). In [Ra2], P. Rabinowitz proved the following theorem by using the Generalized Mountain Pass theorem . If (gl), (g3) hold and a(x) = A, (g2)' g(x, t) = o(ltl) as t --+ O. (g4) tg(x, t) ~ 0 "Ix E 0, t E IR. Then for each A E R, (1.2) possesses a nontrivial solution. Using local linking argument Theorem 1.2 was improved by J. Q. Liu and S. J. Li (see [LiuLil]). The following more general result was given by S. J. Li and M. Willam. Theorem 1.2

Shujie Li

236

Theorem 1.3 [LiWil] . Under the assumptions (gd, (g2)', (g3), if 0 is an eigenvalue of - 6, +a (with Dirichlet boundary condition) , assume also, for some 0 > 0, either

G(x, u)

:=

l"

g(x, s)ds

~ 0 for lui:::; 0,

x

E

n,

or

G(x,u):::; 0 for

lui:::; o, x E n.

Then (1.2) has at least one nontrivila solution. Next we consider the following problem {

- 6, u + a(x)u u=O

= )..g(x, u)

in n on

an.

(1.3),\

H. Brezis and 1. Nirenberg proved the following

Theorem 1.4 (g5)

[BrN1] . Suppose that 9 satisfies (gl), (g2)' and

g(x , t) 0 · 11m - - < . Itl--+oo t If 0 is an eigenvalue of - 6, +a, we assume also that, for some 0 > 0

G(u)

~

0 for

lui:::; o.

Then for every ).. sufficiently large, there are at least two nontrivial solutions of (1.3);,. S. J . Li and M. Willem considered the case G(u) :::; O. Theorem 1.5 [LiWi1] . Under the assumptions of Theorem 1.4. If 0 is an eigenvalue of - 6, +a, we assume also that, for some 0 > 0

G(u) :::; 0 for

lui:::; o.

Then for every ).. sufficiently large, there are at least two nontrivial solutions of (1.3};, . Let X be a Banach space with a direct sum decomposition X = Xl Ef:) X2. The function IE C(X, JR.) has a local linking at 0 with (Xl, X2) if, there exist 0,00 > 0 such that (h) I(u) ~ 0, u E Xl, lIull:::; 0, (12) I(u) ~ 00> 0, u E Xl, lIull = 0, (13) I(u):::; 0, u E X 2, Ilull:::; o.

Some aspects of semilinear elliptic boundary value problem

237

This condition was first introduced and used to find critical points by J . Q. Liu and S. J. Li for the case dimX 2 < 00 (see [LiuLil]). More weak assumption were introduced by E. A. Silva, and used by H. Brezis and L. Nirenberg to prove the existence of a nontrivial critical point, in the case (11), (13) but dim X 2 < 00 (see [Sill and [BrNl]). In [LiWil], S. J . Li and M. Willem droped dimX 2 < 00 and dealt with the case, I is strongly indefinite functional. We shall use the following compactness conditions for the strongly indefinite functionals. Consider two sequences of subspaces

Such that Xj

= U xnj,

J. = 1, 2 .

nEN

For every multi-index a = (aI, (2) E N2 , we denote by XOt the space X~l EBX~2 ' Let us recall that a :::; /3 {:=:::} a1 :::; /31, a2 :::; /32 ' A sequence (an) C N2 is admissible if, for every a E N 2 there is mEN such that n ~ m ===> On ~ a. We denote by lOt the function I restricted to X Ot . Definition. Let c E IR and I E C 1 (X,IR). We say that the function [ satisfies (PS)~ condition if every sequence (UOtJ such that (an) is admissible and UOtn E X Otn , I( u Otn ) -+ c, I~J uOtJ -+ 0 contains a subsequence which converges to a critical point of I. Definition. Let IE C 1 (X, IR). We say that I satisfies the (PS)" condition if every sequence (UOt n ) S.t. (an) is admissible and UOtn E X Otn , sUPnl(uOtJ < 00, I~n (UOt n ) -+ 0, contains a subsequence which converges to a critical point of I. Remark. (PS)" condition was first introduced in [BaBel] . The following theorem we choose from [LiWil]. Theorem 1.6 Suppose IE C 1 (X, IR) satisfies the following assumptions: (Ad I has a local linking at 0 with (Xl, X2) (without (12)), (A 2 ) [satisfies (P S)" condition, (A3) [maps bounded sets into bounded sets, (A4) [is bounded below and d:= infx f < O. Then I has at least two nontrivial critical points. Theorem 1.6 is a more general case of the following theorem given by K. C. Chang.

Shujie Li

238

Theorem 1.7 Let X be a Hilbert space and I E C 2 (X,JR). I satisfies (PS) condition and bounded from below. 0 is not a minmizer of I. If 0 is a nondegenerate critical point with finite Morse index. Then I has at least two nontrivial critical points. Remark. In Theorem 1.6 0 can be degenerate and the Morse index at 0 can be infinity. Now we consider the following problem

{

- 6. u

u=o

= g(x, u) -

h

in 0 on ao

(1.4)

Theorem 1.8 [Cha3]. Suppose that (gl), (g3) hold, and 2 (hr) hE Ln -f'2 (0) is nonnegative, h -:t. o. g(x, t) ~ 0 V(x, t) E 0 X JR1, g(x,O) = 0 and there exists to > 0 such that g(x, to) > o. Then (1.4) has at least two nontrivial solutions. The idea to prove Theorem 1.8 is by using the sub-and super-solution method and Mountain Pass Theorem. Now we consider the multiple solutions of (1.1) . When 9 E C 1 (IT x IRl), it is well known that (1.1) has a positive solution and a negative solution (see [AmRl]) . The following theorem given by Z. Q. Wang point out that (1.1) possesses the third solution. Theorem 1.9 [Wanl]. Suppose that (gr), (g3) hold, and 9 E C 1 (IT X JR1), g(x,O) = gt(x, 0) = o. Then (1.1) has at least three nontrivial solutions. Z. Q. Wang proved this theorem by two methods, one is Morse theory, the other is linking and minimax argument. 1 9 E C (IT x IR.1) can be droped. See recent papers given by J. Q. Liu [Liu2], and S. J. Li -Z. Q. Wang [LiWal]. If we do not require g(x,O) = O. It is possible to discuss the existence of four solutions. The following theorem given by Z. L. Liu and J . X. Sun. Theorem 1.10 Suppose that 9 satisfies (gr), (g3), and (1.1) has a pair of sub-and super-solutions. If there exists m > 0 such that g(x, t) +mt is increase in t. Then (1.1) possesses at least four solutions. Now we state some results for existence of sign--changing solutions. For superlinear case the following theorem was given by T. Bartsch-Z. Q . Wang [BaWal]' and A. Castro-J . Cossio-J. Neuberger [CCN] independently.

Some aspects of semilinear elliptic boundary value problem

239

Theorem 1.11 Suppose that 9 satisfies (g2) , (g3) and (gd' There exist c, 0: > 0 such that 0:

n+2 n-

< --2'

1

9 E C (fl

x JR, JR) .

(g3)' There exists m E JR such that g' (x, t)

> m 'Vt

E JR,

x E fl.

Then (1.1) has at least three nontrivial solutions: one is positive, one is negative and the third is sign-changing. In [LiWa1J, S. J . Li and Z. Q. Wang droped (g3)' and substitute more weak condition (g1) for (gd· Theorem 1.12 [LiWa1) . Suppose that 9 satisfies (gd, (g2), (g3) . Then (1.1) has at least three nontrivial solutions: one is positive, one is negative and the third is sign-changing. In [LiWa1J, a Mountain Pass Theorem in Order Intervals was given in which the position of the mountain pass point is precisely given in term of the order structure. Remark. In [Liu2) J. Q. Liu also gave a proof of Theorem 1.11 without assumption 9 E C 1 . Another simple proof of Theorem 1.11 was given in [LiZha1). Open problem: Does (1.1) have more solutions under the assumptions of Theorem 1.12? Theorem 1.13 Under the assumptions (gd , (g3), if there exist tl 0, t2 > 0 such that g(x , tl) ~ 0, g(x, t2) ~ 0, 'Vx E fl, and 9 satisfies,

(92)'

A2 < go := lim 9(X , t) Itl--tO t


0 such that g(x , t) + mt is increase in t. Then (1.1) has at least seven nontrivial solutions, including two positive solutions , two negative solutions and three sign-changing solutions. This theorem was proved by Dancer and Y. H. Du[DaDu1) using Conley index, by K . C. Chang [Cha4j using Morse theory, and S. J . Li and Z. T . Zhang [LiZha1) using invariance property of flow . Using Mountain Pass Theorem in Order Intervals we can give a very simple proof of Theorem 1.13 under more weak assumptions. Let E be a Hilbert space and PE C E be a closed convex cone. Let X C E be a Banach space which is densely embedded to E . Let P = X n PE and

Shujie Li

240 o

assume that P has nonempty interior p,p 0. We assume any order interval is finitely bounded. eI> is a functional from E to IR and satisfies the following assumptions: (eI>d eI> E CI(E, IR) and satisfies the (PS) condition in E and deformation property in X. eI> only has finitely many isolated critical points. (eI>2) The gradient of eI> is of the form VeI> = Id - KE where KE : E -t E is compact, KE(X) c X, K = KElx -t X is continuous and strongly order preserving, i.e., U > v ===} K(u) » K(v) for all u,v E X, where u » v {:=} o U -

v EP.

(eI>3)

eI> is bounded from below on any order interval in X.

Proposition. [LiWa1). Suppose eI> satisfies (eI>d, (eI>2)' (eI>3), and!! < u is a pair of sub-solution and super-solution of VeI> = 0 in X. Then there exists a negative pseudo gradient flow TJ(t, .) in X such that [!!, u) is positively invariant under the flow and TJ(t,·) points inward on 8[!!,u)\{y,u}. The following is the Mountain Pass Theorem in Order Intervals. Theorem 1.14 [LiWa1) . Suppose eI> satisfies (eI>d, (eI>2)' (eI>3). Suppose there exist four points in X, Vl < V2 , Wl < W2, Vl < W2, [VI , V2) n [WI, W2) = 0 with Vl :S K Vl, V2 > K V2, Wl < K WI, W2 2: K W2. Then eI> has a mountain pass point Uo E [Vl, W2)\ ([Vl, V2) U [Wl, W2)). More precisely, let Vo be the maximal minimizer of eI> in [VI, V2) and Wo be the minimal minimizer of eI> in [Wl, W2), if Vo < Wo then Vo «uo «wo. Moreover Cl(eI>,uo) , the critical group of eI> at Uo is nontrivial. Remark. In Theorem 1.14 one can suppose that VI, W2 are solutions of VeI> = O. Using the Proposition and Theorem 1.14 we have the following Theorem 1.15 [LiWa1) . In Theorem 1.13 if one substitute (gl) for (gd. Then (1.1) has at least two positive solutions ut < ut, at least two negative solutions u > ui, and at least three sign- changing solutions Us, U6, U7 with u < Us < ut , u < U6 < ut where ut , u are the minmizer of the action integral of (1.1), and Us is the mountain pass point. In Theorem 1.13 the condition (96) can be droped. Let

a

a

a

a

E = Ho(o') , X = CJ(O,). (96) guarantees the order preserving property of the negative gradient flow for eI>. Since VeI> = Id - K where K is a compact

241

Some aspects of semilinear elliptic boundary value problem

operator and is strong order preserving. If (g6) does not hold, K may not satisfy the order preserving property. In order to overcome this difficulty we can construct a special vector field which may not be a pseudo gradient vector field. Nevertheless, we can show the flow under this vector field still satisfies the deformation property and prossesses some prescribed invariance property. Let us assume (~2)' ~ E C 2(E, JR) and the gradient of ~ is of the form \7~(u) = AouKE(U) where KE : E ~ E is compact, KE(X) C X and the restriction KE on X, K : X ~ X is of class C 1 and is strongly order preserving, i.e., u > v ==>

»

o

»

v

if

IIAil1(U)II:::; 1

~ E is locally Lipschitz operator such that Ail 1 : E ~ E exists, Ail 1(X) C X and the restriction Ail 1IX : X ~ X is of class C 1 and strongly order preserving. Furthermore, there is M > 0 such that Ao(u) is linear for all u E X and Ilullx :::; M. Finally, we assume that there exists ao > 0 such that

K(u)

K(v) for all u, v E X, where u

IIAil1(u)11

~

aollull

u - v EP; Ao : E

¢:::::}

and

(Ail 1 (u),u) ~ II A il 1 (u)112. In [LiWa1] it was shown that under the assumption (~2)/ one can construct a flow TJ(t, u) along which ~ decreases and satisfies a deformation property, and under which ±P and ±F are both positively invariant, and finally TJ(t, u) leaves any order interval of the form [~, u] invariant provided :!!, u is a pair of sub-and super-solutions of \7~(u) = 0 with 11:!!llx :::; M and Ilullx :::; M. For problem (1.1), let for lui:::; Mo for lui ~ Mo + 1 and m( u) is C 1 and monotonically increasing. Here mo, Mo and p are constants, 0: < p < 2* = ~ ~, 0: was given in (g1). If mo and Mo are large g(x,u) := g(x, u) + m(u) is strictly increasing. Now consider

%

{ ~(u)

-.6. u + m(u)

= g(x,u)

u=o

n

in on 8n

(1.5)

can rewrite

~(u) = .!.

r 1\7 ul dx + inr M(u)dx - inr G(x, u)dx, 2

2 in

where M(u)

= iou m(s)ds,

G(x, u)

= iou g(x, s)ds.

\711>(u) = Ao(u) - K(u)

Then

u E E,

\7~ has a form

Shujie Li

242

where K(u)

= (-6)-l g(X,U) (Ao(u), v)

is strongly order preserving, and

= lt17u \l v + m(u)v)dx

i.e., Ao(u) = u + (-6)-l m(U) . It is easy to check that 4)(u) satisfies (4)2)'. Using the Mountain Pass Theorem in Order Intervals one can prove Theorem 1.16 [LiWaI]. Without condition (g6) Theorem 1.13 still holds. Next we deal with the Dirichlet problem with a changing-sign nonlinearity. Consider in 0 - 6 u = AU + h(x)g(u) (1.6h. { u=o on ao where 0 c ]RN is a bounded open set with smooth boundary, h E G"'(fl) changes sign in 0, a > 0, 9 is a Gl function and satisfies (g2)' and following condition, (g7) For some a, 2 < a < 2*

Suppose that h has a "thick" zero-set (hr) {xlh(x) > o} n {xlh(x)

< o}

=

0.

As in [OuI] one define A*

= inf{11 \l vll~1 Ilvlb = 1,

in

h(x)lvIPdx

= o}.

We introduce the usual action functional

where G(u)

= iou g(s)ds.

We will seek nontrivial critical point of l>,(u) on

HJ(fl) which is equivalent to find weak solution of (1.6}>.. Let Aj(O) be the jth eigenvalue of -6 with Dirichlet boundary data. Let 'Pj be the eigenfunctions corresponding to the eigenvalue Aj. Then Theorem 1.17 [LiWaI]. Assume (g2)', (g7), (hr) hold, and h E G"'(IT) changing sign in o. (1) If A < Al (fl) then (1.6)>. has at least three nontrivial solutions, one is positive, one is negative, and the third is sign-changing.

Some aspects of semilinear elliptic boundary value problem

243

(2) If in addition lim u-+o

for some q

g(u)

lul Q- 2 u

=a>O

(1.7)

> 2, and (1.8)

then there exists X> 0, X~ A. such that if A2(O) < A < X, A ~ a( -.6.), (1.6)A has at least seven nontrivial solutions. More precisely, (2a) (1.6h has at least two positive solutions ut and ut where h(ut) < 0, ut is a local minimizer of JA(u) . (2b) (1.6h has at least two negative solutions u3' and ui where JA(ui) < 0, ui is a local minimizer of JA(u) . (2c) (1.6h has at least three sign-changing solutions U5, U6, U7 where U6 is a mountain pass point of JA(u), J A(U6) < 0, ui < U6 < ut, ui < U7 < ut and U7 is outside [ui, ut] . (3) If in addition (1.7) (1.8) hold, AdO) < A < X, A ~ a( -.6.), then (1.6h has at least five nontrivial solutions. More precisely, (2a) and (2b) hold and there is a sign-changing solution U5. ReInark. In [AIDI] . Alama and Del Pino proved part (1) of Theorem 1.17 except the assertion on the sign-changing solution, and proved part (3) of Theorem 1.17 also except the assertion on the sign-changing solution but for the case Al(O) < A < X < A2(O) . In [BeCDN], Berestycki, Capuzzo Dolcetta and Nirenberg have studied the existence of positive solutions for equations more general than (1.6h.

2. Asymptotically Linear Problems We still consider (1.1) . If there exists goo E IR such that lim Itl-+oo

g(x , t) - goot ·t

=0

'r I ' unllorm y In

X

r.

EH

.

we called g(x, t) is asymptotically linear. Let

f(x, t)

:=

g(x, t) - goot.

Suppose that there exists go E IR such that 9 -

0-

. g(x, t) - g{x, 0) I1m Itl-+o t

uniformly in x E O.

Shujie Li

244

Let

Bo Boo

:= := -

!:::.. u + gou, !:::.. u + gooU'

By mo, moo we denote the dimension of negative subspace of Bo and Boo respectively. By mo, m~ we denote the dimension of nonpositive subspace of Bo and Boo respectively. In [AmaZ1] Amann and Zehnder proved the following Theorem 2.1 Suppose that 9 E C I (IT X RI) , g(x ,O) = 0, go , goo exist, goo (j. 0"( -!:::..) . If moo (j. [mo, mol then (1.1) has at least one nontrivial solution. This theorem was proved by using Conley Index. Later in [Chal], K. C. Chang gave a simple proof by Morse Theory. In [LaSo1] Lazer and Solimini considered more general case. By using the local linking argument , in [LiLiu1J, J. Q. Liu and S. J. Li dealt with the resonant case at zero. Let X = HJ(O)

J(u) = If go

= Ai

~ In I 'V ul 2dx -

In G(u)dx

(2.1)

then we can rewrite (1.1) as following

- !:::.. u + gou = gou + g(x, u) { u=O

m 0 on .80

(2.2)

We make the following assumption (gs)± There exists 6 > 0 such that 1

±(2got2 -

10t

g(x, s)ds) > 0 as

It I ~ 6.

Let X

2

Xl

= {Span{'PI ""

,'Pj-d

Span{'PI, '" ,'Pj}

= Xt-

as (gs)+ holds , as (gs) _ holds,

(in HJ(O)).

One can check that if (gs)± holds, then there exist r > 0,60 > 0 such that

J(u) > 0 as u E Xl, J'(u) 2: 60 > 0 as u E Xl, { J(u) ~ 0 as u E X 2 ,

Ilull ~ r, Ilull = r, lIull ~ r.

(2.3)

Theorem 2.2 [LiLiu1] Under one of the following conditions (1.1) has at least one nontrivial solution.

Some aspects of semilinear elliptic boundary value problem

245

(1) 90 , 900 (j. a( -~), there exists at least one eigenvalue between 900 and 90 ·

(2) 900 (j. a(-~), 90 = >'j E a(-~), (98)± holds and dimX2 i:- moo . Theorem 2.1 and 2.2 are different. But in Theorem 2.2 it is possible to discuss the case moo E [mo, mol provided that (98)± holds. In [LaSol] the resonant case at infinity was discussed Theorem 2.3 Let X be a Hilbert space, X C 2 (X, JR) and satisfies (PS) condition. If

= YEBZ , dim Y

< 00, I(u)

E

inf I(u) = d > -00,

uEZ

I(u) ~

-00

as

lIull ~ 00

u E Y.

Suppose () is a isolated critical point of I. D2I(()) is the Fredholm operator. Either dim Y > mo or dim Y < mo . Then I has a nontrivial critical point u* . In [BaLil] T . Bartsch and S. J. Li got more information. Theorem 2.4 Then

Under the assumptions of Theorem 2.3, let dim Y

= k.

where C k (1,00) is the k-th critical group of I at infinity, defined by Ck(I, 00) := Hk(X , Ia) , where Ia = {x E XII(x) ~ a}, a < infuEK I(u), K is the critical point set of I , and Hk(X, Ia) is the singular k-relative homology group. In [LiWil] S. J. Li and M. Willem substituted following (98)± for (98)± in Theorem 2.2, (98)± There exists 8

> 0 such that ±(i90t2

-lot

9(X , s)ds) 2: 0 for

It I ~ 8.

Concerning the multiple solutions, in [ChaLL1], K. C. Chang, S. J. Li and J . Q. Liu Proved the following

=

Theorem 2.5 Suppose that 9 E Cl(IT X JRl), 9(X,O) 0 , 90 < >'1,900 > >'2' Then under the one of the following conditions (1.1) has at least three

nontrivial solutions (including one positive, one negative) . (1) 900 (j. a( -~), (2) 900 E a( -~), ¢(x, t) := 9(X, t) - 900t is bounded and satisfies the Landesman-Lazer condition

Shujie Li

246

where (x, t)

= lot ¢(x, s)ds,

Span{CP1, " ', CPm}

= Ker( -

6. -goe'!).

Theorem 2.6 Suppose that 9 E C 1(n x JR1), g(x,O) = 0, Al < go < Ak < goo, goo satisfies assumption (1) or (2) in Theorem 2.5. If there exists to # such that g(x, to) = \Ix E Then (1.1) has at least three nontrivial solutions. If A2 < go < Ak < goo \Ix E Then (1.1) has at least four nontrivial solutions. Similar result also was given by Dancer and Y. H. Du (see [DaDul]) . In [BaWal] T . Bartsch and Z. Q. Wang pointed out that the third solution of Theorem 2.5 is sign--changing.

°

°

n.

n.

Theorem 2.7 Suppose that 9 E C 1(n x JR1), g(x,O) = O,go < A1,goo > A2, goo satisfies the assumption (1) or (2) in Theorem 2.5. Then (1.1) has a sign--changing solution. Remark. Using the peseudo gradient flow introduced in [LiWal) and combining the sub-and super-solution argument the assumption 9 E C 1 in Theorem 2.5, 2.6 and 2.7 can be droped. The following theorem given by T . Bartsch, K. C. Chang and Z. Q. Wang. Theorem 2.8

[BaCWl). Under the assumptions of Theorem 2.7 if 9 also

(t'

satisfies g' (x, t) 2:: 9 t) \Ix En, t E lit Then (1.1) has one more sign--changing solution with Morse index 2. S. J. Li and Z. T . Zhang have the following Theorem 2.9 [LiZha2] . If 9 E C 1(n x JR), g(x,O) = 0, go < A1,goo = Ak, k 2:: 2, g' (x, t) ::; r < Ak+1 . If g(x, t) - goot satisfies assumption (2) in Theorem 2.5. Then (1.1) has at least four nontrivial solutions: one is positive, one is negative, one is sign-changing. As we stated above if the nonlinearity crosses at least one eigenvalue (1.1) has at least one nontrivial solution. For the case in which the nonlinearity crosses the first eigenvalue, the existence of multiple solutions have been studied by many authors see [AhLPl), [AmI), [AmMl), [Danl), [Hi 1), [HoI), [Stl], [Til]. For the case in which the nonlinearity crosses the higher eigenvalues, if there are always two nontrivial solutions it is still an open problem. Under some additional condition, in [LiWi2), S. J. Li and M. Willem proved the following Theorem 2.10 Suppose that go ¢ eT( -6.), Aj < goo < Aj+1 < go, g'(x, t) E JR, j > 1. Then (1.1) has at least two nontrivial solutions.

C> Aj \Ix E n u

~

Some a.spects of semilinear elliptic boundary value problem

247

Theorem 2.11 Suppose that Aj < go < Aj+1 ::; Ak < goo < Ak+1, g'(x, t) ::; c < Ak+1, Vx E n, t E R Then (1.1) has at least two nontrivial solutions. It is similar to the superlinear problem. Using the Mountain Pass Theorem in Order Intervals we can prove

Theorem 2.12 Suppose that go > A2, go f/. CT( -6), goo ~ A2. If goo E CT( -6) also require the assumption (2) of Theorem 2.5. If there exist t 1, t2 E JR1, t1 < 0, t2 > 0 such that g(x, td ~ 0, g(x, t2) ::; 0 Vx E n. Then (1.1) has at lea.st two positive solutions ut < ut, two negative solutions u3" > ui and three sign--changing solutions us, U6, U7. Where ut, u3" are the minimizers of action functional I (u), ut, ui , Us are the mountain pass points of I (u). Remark. We do not ask any smooth assumption on g. For the case g E C 1(IT x JR) the related results see [Cha5], [DaDu1] and [LiZha1]. Corollary 2.13 In theorem 2.12 if A1 < goo < A2, then (1.1) has two positive solutions, two negative solutions and two sign-changing solutions. If A1 < go < A2 then (1.1) has two positive solutions, two negative solutions and one sign-changing solutions. If A1 < go < A2, A1 < goo < A2 then (1.1) has two positive solutions and two negative solutions. The idea to prove Theorem 2.12 and Corollary 2.13 was given in [LiWa1], [LiZha1]. Now we are in a position to consider the resonant problem. If go E CT( -6) or goo E CT( - 6.) we call (1.1) is resonant at origen (or at infinity) respectively. It is more delicate to compute the critical groups at degenerate critical points. Let I(u) = ~ I '1 ul 2 - ~ goou 2 + ~gooU2 - G(u),

In

In

In

~(t) =

(x, t) = g(x, t) - Akt,

lot (x, s)ds.

In [Cha4] K. C. Chang pointed out Theorem 2.14

In ~(uo)dx

Suppose that goo -t

= Ak,

(x, t) is a bounded function and

+00 as Uo E Ker (- 6 -Ak) and Iluoll

Then [ satisfies the (P S) condition and

-t

00.

Shujie Li

248 where r

= max {dimensions of nonpositive eigenspaces of

- 6. -

Ad

Remark. In [AhLP1) Ahmod, Lazer and Paul proved that under the assumptions of Theorem 2.14 problem (1.1) has a solution. In fact, using the Saddle Point Theorem it is easy to prove the result. Theorem 2.14 give more information: (1.1) has a solution u* and Cr(I, u*) -:j; O. In [Si1) and [Col) E. Silva and D. Costa generalized the result given in [AhLP1) and they pointed out that if

and

. hm

Itl~oo

~(x, t)

-2-

It I

= +00(-00).

Q:

Then (1.1) has a solution. Remark.

in ~(Uo)

In Theorem 2.4

dx -t +00.

Then

if one substitute

L~(uo)dx

-t -00

for

Cq(I,oo) = Oq,mooG where moo = max {

dimensions of negative eigenspace of - 6. - Ad, see [LiLiu2). In [BaLil) T. Bartsch and S. J. Li droped the boundedness of 0 and c: E (0,1) such that (I'(u),v) ::; 0 for any u = v + wE V EB W with lIull ~ Rand Ilwll ::; c:llull.

Some aspects of semilinear elliptic boundary value problem

249

Now we apply Theorem 2.15 to (1.1) . Let ¢(x, t) = g(x, t) - goot and ¢ satisfies the following condition. (¢~) There exist 0, let

Sr = R/(rZ) and Er = Wl,2(SnR2n) with the usual norm

Then Er is a Hilbert space. We denote by (., ·)r the corresponding inner product in E r . Now we define the 'odd' subspace E~dd of Er by E~dd

= {u E Er Iu(r/2 + t) = -u(t)}.

It is clear that E~dd is also a Hilbert space with equivalent norm

By (V5), the function V is even, we define a functional

f on

E~dd

by

(3.1)

Iteration formula for the w index with applications

263

It is clear that any critical point x of f in E~dd is a solution of the system (1.2) with x(T/2 + t) = -x(t) . If x E E~dd is a critical point of f in E~dd, then the following quadratical form defined in E~dd by (3.2)

has finite Morse index and nullity which are defined as the dimension of the greatest negative definite subspace and the null subspace, respectively. By (V5), A(t) := V"(x(t)) is T/2-periodic, we can see that there holds

r/ (lul 2

'l/Jr(u,u)

=2

10

2

-

(V"(x(t))u,u))dt, Vu E E r / 2,-1,

(3.3)

where E r / 2,-1 = {u E Wl,2([o,T],Rn) I u(T/2) = -u(O)}, thus by the arguments of [Lol], we know that in fact the Morse index pair (m-(x), v(x)) is exactly the w (w = -1) index (L 1,r/2(X), V-l,r/2(X)) of x as a solution of the second Hamiltonian system which correspending to a linear first Haimiltonian system with coefficent matrix B(t) =

(~ ~(t))'

By the mountain pass lemma, we have the following result. Theorem 3.1. Suppose the function V satisfies the conditions (Vl)-(V5), then for every T > 0, there is a critical point x of the functional f in E~dd with its Morse index satisfying

(3.4) If the function V satisfies the conditions (Vl)-(V3), (V5)-(V6), then for every there is a critical point x of the functional f in E~dd with its Morse index satisfying

o < T < 27r /,;w,

(3.5) Proof. We only prove the first result, the second result is similar to prove. It is easy to check that the function f in E~dd satisfies all conditions of the mountain pass lemma. In fact, by the condition (V4), it is easy to see that there exist p > 0 and a > 0 such that f(u) ~ a

> 0,

Vu E E~dd

n Bp(O).

On other hand, f(O) = 0 and by (V2), there exists R > p and Uo E E~dd with Iluoll ~ R such that f(uo) ~ O. Thus by mountain pass lemma, there exists a critical point x with its morse index m - (x) ~ 1.

o

Chun-gen Liu

264

If x E E~dd is a non-zero solution of the system (1.2), set y(t) = x(t) and = (x(t),y(t))T, then z satisfies z(7/2+t) = -z(t) for all t E R and it is a non-zero 7-periodic solution of the following first order Hamiltonian system

z (t)

z(t) where H(z)

= 1/21Y12 + V(x).

= JH'( z (t))

(3.6)

From (3.6), there holds

z.. - JH"( z )z.. Thus

z is a non-zero solution of the following linearized equation u = JH"( z )u, u E R2n .

(3.7)

Let TZ(t) be the fundamental solution starting from the identity matrix hn . Then -1 E a(')'z(7/2)), thus we have

(3.8) On other hand , if x E E~dd is a non-zero solution of the system (1.2), then it is not even times iteration of any periodic solution of (1.2) . i.e., if x is a 7/k-periodic solution of (1.2), then k is odd. Further more, x E E~1~, i.e., x(7/(2k) + t) = -x(t) . In fact, suppose k = 2m - 1, mEN, then there holds 7

x(2(2m _ 1)

+ t)

7 = x( 2(2m _ 1)

(m - 1)7 1)

+ 2(2m _

+ t)

7 = x("2

+ t)

= -x(t). (3.9)

4. Iteration inequality of the symmetric Morse index theory and the minimal periodic problems Let x E E~dd be a critical point of the functional f on E~dd, and A(t) = V"(x(t)). By the Proposition 2.3 of [Lo1], f(x) defines the following bilinear form on E~dd

1/Jr(x,y)

r (x ·iJ-A(t)x·y)dt = ior/ (x ·iJ-A(t)x · y)dt,

= io

2

2

(4.1) We denote the Morse index of'l/J on E~dd by (L 1,r/2(X),1I-1 ,r/2(X)) . For odd number k E 2N - 1, by the arguments in the end of the section 3, the k-times iteration of the solution x is a kT-periodic solution of (1.2) satisfies x(kT /2 + t) = -x(t) . So we can define the iterated index (i-l,kT/2(X),1I-1,kT/2(X)). Estimating the iterated index is crucial in our work here.

Iteration f(j)Fmula for the w index with applications

265

We note that if setting u(t) = x(t) and v(t) = y(t) in (4.1), then x(t) = (II T u)(t) and y(t) = (II T v)(t), the operator lIT : L~ --+ W 1,2([0, T], Rn) is bounded and defined in [Ek] . Thus we can define the following quadratic form in L~ = L2([0, T], R2n) (4 .2)

Defining an operator AT : L~ --+ L~ by

there holds (4.3)

where we have denoted £T = I - AT and (,,·h the L2 inner product. For the iterated symmetric Morse index, we have following estimate. Theorem 4.1. Suppose A(t) ~ O. For k E 2N - 1, there holds . k-l(. Ll,kT/2 ~ -2- Ll,T/2

+ /J-l,T/2 ) .

(4.4)

Proof. The case k = 1 is trivial. In the following we suppose k > 1. Firstly we note that the Morse indices of ¢>T in L~ equal to the Morse indices of 'l/JT in the space E~dd, and there is a ¢>T-orthogonal splitting

with: dimL_

= L 1 ,T/2 and ¢>T(U,U) < 0 Vu E L_ \ {O} dimL o = /J-l,T/2 and Lo = ker¢>T'

For 0 ~ j ~ (k - 1)/2 and k > 1, we define an operator T j

Tju(t) = {

:

U(t-jT) jT~t~jT+T/2 . - (k-l)T)' kT < t < . U(t, - JT 2 JT + "2 _ _ JT o otherwise

Now define subspaces M j and N j of L~T by

L~ --+ L~T by

+ (k+l)T 2

Chun-gen Liu

266 The subspaces are mutually orthogonal. Setting M = ffiMj and

N = ffiN j , j

j

we check easily that 0 being sufficiently small. We also define Sp(2n)~ = Sp(2n) \ Sp(2n)~. For any two continuous arcs ~ and 11 : [0,7] -+ Sp(2n) with ~(7) = 11(0), we define as usual:

11 * ~(t) = {

~(2t),

11(2t - 7),

if 0 ::; t ::; 7/2, if 7/2 ::; t ::; 7.

We define a special path ~n E P T (2n) by ~n(t)

. t t t)_l , ... , (2 - -t)_l) , = dlag(2 - -, ... ,2 - -, (2 - 7

7

7

for 0 ::; t ::;

7.

7

Definition 4.1 Let wE U. For any M E Sp(2n) , we define

lIw(M) = dime kerc(M - wI) . For any

7

>

(4.1)

0 and, E PT(2n), we define

lIw(r) = lIw(r(7)).

(4.2)

If ,(7) E Sp(2n)~, we say, is w-nondegenerate and define

iw(r) = [Sp(2n)~ :, * ~nJ,

(4.3)

278

Yiming Long

where the right hand side of (4 .3) is the usual homotopy intersection number, and the orientation of'Y * ~n is its positive time direction under homotopy with fixed end points. If 'Y(r) E Sp(2n)~, we define (3 E P,.(2n) is w-nondegenerate

(4.4)

and CO -sufficiently close to 'Y}' Then (iwb),vwb)) E Z x {0,1, .. . ,2n},

is called the index function of'Y at w . Note that the right hand side of (4.4) is always finite as proved in [29] or [30] respectively. For any symplectic path 'Y E p,.(2n) and mEN, we define its m-th iteration 'Y m : [0, mr] -+ Sp(2n) by 'Ym(t)

= 'Y(t -

'Vjr ~ t ~ (j

jrh(r)j,

+ l)r, j = 0,1, ... , m

- 1.

We denote the extended path on [0, +00) still by 'Y. Fix a ~ E 1-l(2n) and a real number Q E (1,2). For any (r,x) E and mEN, we define its m-th iteration xm : R/(mrZ) -+ R 2 n by 'Vjr

~

t

~

(j

+ l)r,

j = 0,1,2, . .. , m - 1.

(4.5)

J(~,Q)

(4.6)

We still denote by x its extension to [0, +00). Definition 4.2 For any 'Y E p,.(2n), we define

'Vm E N.

(4.7)

The mean index ib, m) per mr for mEN is defined by

~ ( 'Y, m )

Z

=

I'

1m k-++oo

ib, mk) . k

(4.8)

For any M E Sp(2n) and wE U, we define the splitting numbers str(w) of M at w by str(w) = lim iw exp(±.;=T. defined by (1.3).

i(t) { z(I)

=

=

JH~(z (t)) ,

"it E R,

z(O).

Define

E = {u E L(e>.-l)!e>.(R/Z, R2n) I

10

(4.13)

1

udt = O}.

(4.14)

The corresponding Clarke-Ekeland dual action functional f : E --+ R is defined by 1 1 (4.15) f(u) = {-(Ju, TIu) + H~( -Jundt, o 2 and f E C 2(E,R). Here TIu is defined by ftTIu = u and fo1 TIudt = 0, and the

1

usual dual function H; of He>. is defined by H~(x) =

sup ((x , y) - He>. (y)) .

(4.16)

yER2n

Here (' , .) denotes the standard inner product of R 2n. Following §V.3 of [10], we denote by "ind" the Fadell-Rabinowitz Sl-action cohomology index theory for Sl-invariant subsets of E defined in [10] (cf. also [14] of E. Fadell and P. Rabinowitz for the original definition) . For [f]e == {u E Elf (u) ~ c}, the following critical values of f are defined ck=inf{c

i(x, 1) - n

+ 1,

.( x,l ) - eb",(r)) 2 +1

~ Z

"1m EN.

(4.33)

Note that the proof of Theorem 4.1 depends on the Bott-type iteration Formula of the Maslov-type index theory and the complete understanding of the splitting numbers established in [29]. The proof of Theorem 4.2 depends on the precise iteration formulae of the Maslov-type index theory established in [31]. This proof in fact gives a universal method to detect and prove any iteration inequalities of the Maslov-type index theory. Theorem 4.2 guarantees the index intervals and index jumps lining up nicely. By Theorem 4.1, the change of i(x, m) in m consists of a linearly increasing term m(ib, 1) + St(l) - C(M)), rotator terms E(,;:) with SM(eV-1S) > 0, and a bounded term. Then the control of the location and the size of the index jumps 92m;-1(rj,Xj) for 1 ~ j ~ q mainly depends on the control of all the rotators in terms of the iteration time 2mj - l's for 1 ~ j ~ q so that they get the largest jump simultaneously. Then the problem is reduced to the study of dynamics on tori . By our study on closed additive subgroups of the standard tori, the proof of Theorem 3.1 is completed. The proofs of Theorems 3.2 and 3.3 follow from the complete algebraic understanding of the splitting numbers given by Theorem 4.11 of [29]. Given a hypersurface ~ E 1is(2n), the solution orbits of (1.1) are devided into two sets, symmetric ones and non-symmetric ones. A symmetric orbit has Maslov-type index increasing faster than the usual ones, which allows more effective integers corresponding to such orbits located in the interval on the left hand side of (4.28) . On the other hand, non-symmetric orbits always appears as pairs {[(r, x)], [(r, -x)]), and both (r, x) and (r, -x) possess the same index, nullity, and splitting numbers. Then in this case one effective integer in the interval on the left hand side of (4.28) corresponds to two different orbits [(r, x)] and [(r, -x)]. In such a way, Theorem 3.4 is proved.

Dynamics on compact convex hypersurfaces in R 2n

283

Our main idea in the proof of Theorem 3.5 is to show the existence of one closed characteristic [(Tj,X j )) found by Theorem 3.1 which makes both equalities hold in (4.33) for the chosen iteration time m = 2mj. Then it must be elliptic. This closed characteristic is minimal in a certain sense. The proof of Theorem 3.6 depends on the understanding of the mean index sequence of iterations of closed characteristics. When ~ E 1i-(2n), the corresponding mean indices of iterations of closed characteristics must strictly increase suitably. Then we prove that if any two of the solution orbits found by Theorem 3.1 are rational, by our choice of the iteration times, the two corresponding mean indices must be equal to each other, and then yields a contradiction. The proof of Theorem 3.7 depends on the precise iteration formulae of the Maslov-type index theory established in [31). By these formulae , all the closed characteristics on ~ can be classified into finitely many families according to their different index iteration patterns. Then in each case we prove that (4.25) can not hold if there exist fewer than two elliptic closed characteristics. To prove Theorem 3.8, we further observe that the elliptic solution found in the Theorem 3.5 corresponds to a vertex X of some cube [0, l)k for some kEN. By the proof of Theorem 3.6, when n 2:: 2 we can find two such vertexes, and then we prove that they produce two different elliptic closed orbits.

5

Further considerations and open problems

The aim of the study on the problem (1.1) is three folds . The first is to completely understand the structure of the set of closed characteristics on any given hypersurface ~ E 1i(2n), and then to further understand the characteristic flow on ~. The second is to understand the structure of the set 1i(2n) . The third is to use it as the most typical example in the study of the closed solution orbits of the Reeb vector fields on general contact manifold. Our recent results give answers to these problems at different degrees. Based on our Theorems 3.1 to 3.8 and earlier results mentioned in the above section 2, it is natural to propose the following conjectures. Conjecture 1 There holds

{# j(~) I ~ E 1i(2n)} = {k E N I [~) + 1 ~ k ~ n} U {+oo} .

(5 .1)

This conjecture actually includes three steps. The first step is to construct (or prove the existence of) a particular hypersurface ~ E 1i(2n) such that # j(~) = [~) + 1. The second step is a generalization of the result (2.8) of Hofer-Wysocki-Zehnder to higher dimensions , i.e. # j(~)

2:: n + 1 implies

# j(~) =

+00,

for

~

E 1i(2n).

(5.2)

Yiming Long

284

The third step is to study how the # j(,£) decreases from n to [n/2] + 1 when '£, for example the weakly non-resonant ellipsoid, loses its symmetry. Conjecture 2. For any '£ E 1l-(2n) , all the closed characteristics on '£ are elliptic. Here we would also like to remind the readers on the following reduced version of the conjecture (2.10) based on our Theorem 3.5: Conjecture 3. # je('£) 2:: 1 for any '£ E 1l00(2n). More generally, it is interesting to know the global structure of the set j(,£) for any '£ E 1l 00 (2n). It is a interesting problem to clarify the relation between the contact (or corresponding symplectic) invariants of the given contact manifold and the number of periodic orbits of the Reeb vector field on it. Such a relation should include the case of hypersurfaces in 1l(2n), specially our Theorems 3.1-3.8. It is interesting to know the relation between our (In('£) and the possible topological invariants. For this problem, we refer the readers to the recent result [22] for an exciting work. As pointed out in [31], the study of the stability of closed characteristics can be reduced to the following conjectures on primes in the number theory. Given any pEN, we define an integer pair set Y(p) = {(pn - 1,pn + 1) In EN}.

(5.3)

Conjecture 4. For any pEN. the set Y(p) contains infinitely many prime number pairs. Note that this is a slight general version of the conjecture on twin primes. Given any cp E [0,1] \ Q, we define an integer set Z(cp)

=N

\ {2n+ 2[ncp]

+ lin EN}.

(5.4)

Conjecture 5. For any cp E [0,1] \ Q, the set Z(cp) contains infinitely many prime numbers. It is even more interesting if one can give dynamical system proofs to these conjectures on prime numbers.

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286

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[16] J. Han and Y. Long, Normal forms of symplectic matrices (II) . Nankai Inst. of Math. Nankai Univ. Preprint. (1997). Acta Sci. Natur. Univ. Nankaiensis. 32 (1999) 30-41. [17] H. Hofer, A new proof of a result of Ekeland and Lasry concerning the number of periodic Hamiltonian trajectories on a prescribed energy surfaces. Boll. UMI. 6 (1982) 931-942. [18] H. Hofer, K. Wysocki, and E . Zehnder, The dynamics on three-dimensional strictly convex energy surfaces. Ann. of Math . 148 (1998) 197-289. [19] V. J . Horn, Beitriige zur Theorie der kleinen Schwingungen. Zeit. Math. Phys . 48 (1903) 400-434. [20] A. Liapunov, ProbIeme general de la stabilite du mouvement. Russian edition (1892), Ann. Fac. Sci. Toulouse 9 (1907) 203-474. [21] C. Liu, Y. Long, and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in R2n. Nankai Inst. Math. Preprint No. 1999-M-003. Submitted. [22] G. Liu and G. Tian, Weinstein conjecture and GW invariants. Preprint. (1997) [23] Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Science in China (Scientia Sinica) . Series A. 7. (1990) . 673-682 . (Chinese edition), 33. (1990) . 1409-1419. (English edition). [24] Y. Long, The structure of the singular symplectic matrix set. Science in China (Scientia Sinica) . Series A. 5. (1991) . 457-465. (Chinese edition), 34. (1991). 897-907. (English edition). [25] Y. Long, The Index Theory of Hamiltonian Systems with Applications. (In Chinese). Science Press. Beijing. (1993) . [26] Y. Long, A Maslov-type index theory for symplectic paths. Top. Meth. Nonl. Anal. 10 (1997) 47-78. [27] Y. Long, Hyperbolic closed characteristics on compact convex smooth hypersurfaces in R2n . Nankai Inst. of Math. Nankai Univ. Preprint. (1996) . J. Diff. Equa. 150 (1998) , 227-249. [28] Y. Long, The topological structures of w-subsets of symplectic groups. Nankai Inst. of Math. Nankai Univ. Preprint. (1995). Acta Math. Sinica. English Series. 15. (1999) 255-268.

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[29] Y. Long, Bott formula of the Maslov-type index theory. Nankai Inst. of Math. Nankai Univ. Preprint. (1995, Revised 1996, 1997) . Pacific J. Math . 187 (1999), 113-149. [30] Y. Long, The Maslov-type index and its iteration theory with applications to Hamiltonian systems. Third School on Nonlinear Analysis and Applications to Differential Equations. (10. 12-30,1998). ICTP Lecture Notes. SMR 1071/2 . . Proc. of Inst. of Math . Academia Sinica. to appear. [31] Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. ICTP Preprint (1998). Nankai Inst. Math. Preprint No. 1999-M-001. Advances in Math. to appear. [32] Y. Long and D. Dong, Normal forms of symplectic matrices . Nankai Inst. of Math. Nankai Univ. Preprint . (1995) . Acta Math . Sinica. to appear. [33] Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In Stoc. Proc. Phys. and Geom. S. Albeverio et al. ed. World Sci. (1990) . 528-563. [34] Y. Long and C. Zhu, Maslov-type index theory for symplectic paths and spectral flow (II). Nankai Inst. of Math. Nankai Univ. Preprint. (1997) . Revised 1998. Chinese Ann. of Math. 21B :1(2000), 89-108. [35] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in R2n . Nankai Inst. Math . Preprint No. 1999-M-002. Submitted. [36] J . K. Moser, Periodic orbits near an equilibrium and a theorem by A. Weinstein. Comm. Pure Appl. Math. 29 (1976), 727-747. [37] P. Rabinowitz, Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31. (1978) . 157-184. [38] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conf. Ser. in Math . no.65. Amer. Math. Soc. (1986). [39] A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems. Bull. Soc. Math. France. 116 (1988) , 171-197. [40) C. Viterbo, A new obstruction to embedding Lagrangian tori. Invent. Math. 100. (1990) 301-320. [41] A. Weinstein, Normal modes for nonlinear Hamiltonian systems. Inven . Math. 20. (1973).47-57.

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[42] A. Weinstein, Periodic orbits for convex Hamiltonian systems. Ann. of Math. 108. (1978). 507-518. [43] V. Yakubovich and V. Starzhinskii, Linear Differential Equations with Periodic Coefficients. New York, John Wiley & Sons. (1975). [44] C. Zhu and Y. Long, Maslov-type index theory for symplectic paths and spectral flow (l). Nankai Inst. of Math. Nankai Univ. Preprint. (1997) . Revised 1998. Chinese Ann. of Math. 20 B. (1999) 413-424.

Gromov-Witten invariants on compact symplectic manifolds with contact type boundaries and applications Guangcun LuI f~anKaI

!nStltute ot MatnematIcs, fllanKaI umverslty Tianjin 300071, P. R. China (E-mail: [email protected])

Abstract In this note we define the Gromov-Witten invariants on any compact symplectic manifolds with contact type boundaries and outline an application of it to the topological rigidity of Hamiltonian loops on this class of manifolds.

In the past years the Gromov-Witten invariant theory on the closed symplectic manifolds was established and studied well(cf.[Gr][Wl][W2][Rl][RT][KM] [McSal][K][FO][LT2][R2][Sie]). Clearly, it is natural and necessary to establish this theory on the symplectic manifolds which are noncompact or with boundaries in both the theory and the application. In [Lu] author defined the Gromov-Witten invariants on a class of noncom pact symplectic manifolds. As a special case the Gromov-Witten invariants on any weakly monotone compact symplectic manifolds with contact type boundaries were given. In addition we also studied topological rigidity of Hamiltonian loops with compact support that is a generalization of results in [LMP] . In this note we shall define the Gromov-Witten invariants on any compact symplectic manifolds with contact type boundaries and outline the general versions of Corollary 6.3 and 6.17 in [Lu].

1

J -convexity and hypersurfaces of contact type

In this section we will review some special important notions. Let (M, J) be an almost complex manifold of dimension 2n and SCM a smooth compact 1 Partially

supported by the NNSF of China.

289

Guangcun Lu

290

e

connected oriented hypersurface. J determines a complex vector bundle := T S n J(T S) over S of rank n - 1, which is a real vector subbundle of T S. For a xES we take a base Ul," ' , Un, JxUl,"', Jxu n for {~, and v E TxS such that

form a base for TxS and give the specified orientation of S . We call the part directed by Jxv the "interior' of S and one directed by -Jxv the "exterior' of S . It is easily proved that there exists a smooth nowhere vanishing I-form a on S such that a(v)

> 0 and

{~

= Ker(a x ).

Such a I-form is called the defining I-form of the sub bundle e. Moreover, one can prove that the defining I-form of S is unique up to the multiplication by a positive smooth function factor . Definition 1.1 Let a be a defining I-form of S, if da(v, Jv) > 0 for every nonzero vector vEe, S is said to be J -convex. Similarly, S is called J -concave ifda(v,Jv)

< 0 for all nonzero vectors vEe.

It is easily proved that this definition is independent of the choice of the

defining I-forms. Notice that there always exist an open neighborhood U of S and a smooth function p : U -+ lR such that 0 E lR is an regular value of it, S = p-l(O) and {x E U I p(x) case one can easily show that

< O} is contained in the interior of S. In this

(1.1)

a := -(J*dp)ls

e.

is a defining I-form of By the definition it is not difficult to prove that S is also J'-convex with respect to any almost complex structure J' on M such

e e'

that = and Jle = J'le" The following property is the most important one for J-convex surface. Lemma 1.2([E][Mc1]) Let S be a J-convex hypersuface in an almost comThen any J-

plex manifold (M, w) and W the union of it and its interior.

holomorphic curve contained in W can not touch S at an interior regular point. Proof Let ~ be a compact Riemann surface and U : ~ -+ M a nonconstant J-holomorphic curve with u(~) C W . We want to prove U({ z E Int(~) I du(z) =F O}) n S

= 0.

Take an open neighborhood U of S and a smooth function p : U -+ lR as above. Then (1.1) gives a defining I-form of Choose a € > 0 so small that [-€,€] only contains the regular values of p and p-l ([ - €, €]) C U.

e.

291

Gromov-Witten invariants on compact symplectic manifolds

Assume that there exists Zo E Int(~), du(zo) i:- 0 such that u(zo) E S. Since the critical points of J-holomorphic curves are isolated, by choosing a local coordinate chart we may assume that there exists a J-holomorphic disk v : D = {I z I ~ I} -+ M such that (1.2)

v(O) =

XES,

i:- 0 Vz ED.

v(D) C p-l ([ - f , 0]), dv(z)

Writting z = s + it, since I := po v : D -+ ~ attains its maximum at z = 0 we get dp(x)(8s l(0» = dp(x)(8d(0» = O. and thus

8 sl(0), 8d(0) E TxS \ {O}.

(1.3)

But 8 s 1 + J(v)8d = 0 we obtain that both 8 s l(0) and 8d(0) belong to ~;. Denote by (3 := -J*dp. This is a smooth I-form on an open neighborhood U of Sand (3ls = a. From the J-convexity of S it follows that

d(3(v(0»(8s l(0), J(v(0»8 s l(0» = da(v(0»)(8 s l(0), J(v(0»8 s l(0» > O. Thus for sufficiently small 15 > 0 it holds that

d(3(v(z»)(8sJ(z) , J(v(z»8 s l(z» > 0, Vz E Dli := {Izl ~ 15}.

(1.4)

Let dC

= Jod,

where Jo denotes the standard complex structure on C. By the

definition

dd C 1= -6.(I)ds 1\ dt on Dli·

(1.5)

o

Since dC I = JodI = Jodp 0 dv = J 0 v*(dp) = J (J 0 v*)*dp = v· 0 J*dp = -v*(3 we arrive at

dd C 1= d(v*

0

o

0

(v*)*dp = (v*

0

Jo)*dp

=

J*dp) = -v*d(3 = -d(3(8s v, 8t v)ds 1\ dt.

Hence this and (1.4) together lead to (1.6)

6.(1) = d(3(8s v,8t v) = d(3(8s v,J(v)8s v) > 0 on Dli ·

But 118 D 6 ~ O. The Maximum principle leads to p(v(O» < O. This contradicts to x = v(O) E S.

Ilrnt(D6)

< O. Specially, 0

Guangcun Lu

292

Remark 1.3 In the past arguments, for example, Lemma 1.4 in [Mel), they, in fact, only considers the case that 0 is a regular point, and neglects that of 0 being a critical point. From the above proof we may know that in the case of

8s v(O) = 8t v(O) = 0 it is impossible to use this method to get the conclusion that x = v(O) ~ S. However, later we will show that there exists another almost = and such that not only S is complex structure J' near S for which J'-convex but also every J'-holomorphic curve contained in W cannot touch S from the inside.

e e'

A class of important pseudo-convex hypersurface comes from the hypersurface of contact type([We)). Recall that a compact smooth hypersurface S in a symplectic manifold (M,w) is said to have contact type if there exists a smooth vector field X in a neighborhood U of S that is transverse to S and Lxw = do ixw

+ ix

0

dw = w. An equivalient statement is that there exists

a contact form 0 on S such that do = wis. Define the part directed by X the outside of S. It is easily checked that this definition is independent of the choice of such X and o. Thus different choices of X may give the same orientation on S by ixw n . Later, the orientation on S always means this one without special statements. The following lemma gives the relation between J-convex hypersurface and one of contact type. Lemma 1.4 For every hypersurface S of contact type in a symplectic manifold (M,w) there exists a w-compatible almost complex structure J E .J(M,w) such that S is not only J -convex with respect to the orientation just defined but also has the property:

(1.7) every J-holomorphic curve in the interior of S cannot touch it from the inside. Conversely, if an oriented hypersuface S C (M, J) is a J -convex there always exists a contact form on S. Proof The second claim is clear. In fact, let p : U -+ IR give a defining I-form

e

e

-(J*dp)ls of := TSn (JTS) as above. Then = Kera is a sub bundle of T S of rank 2n - 2, and do( v, Jv) > 0 for any nonzero vector vEe because Hence of J -convexity of S . The latter implies that do is nondegenerate on o is a contact form on Sand o2n-1 gives the specified orientation of S.

0=

e.

For the first claim Proposition 8.14 in [ABKLR) showed that there exists a E .J(M,w) such that S is J-convex. The remainder is to find such a J

Gromov-Witten invariants on compact symplectic manifolds

293

also satisfying (1.7). To this goal let us consider the symplectizatins of the contact manifold (S,a), (N,w) := (IR x S,d(eta» . The usual implict function theorem arguments show that there exist a € > and a diffeomorphism from (-2€,2€) x S onto an open neighborhood U C p-1([-2€,2€)) of Sin M,

°

(1.8)

~: (-2€, 2€)

x S -t U, ~(O,x)

= x,

"Ix E S,

that maps (-2€,O) x S into the inside of S( in fact, p-1([-2€,0))). Denote by XO/ the Reeb vector field determined by ixaa = 1 and iXada = 0. Then one gets a natural splitting TS = IRXO/ EB Let 7r0/ : TS -t be the projection along IRXO/ given by v t-+ v - a(v)XO/. These may pick out a class of important almost complex structures in .J(N,O) as follows:

e.

(1.9)

J(t, x)(h, k)

e

= (-a(x)(k), J(x) (7rO/k) + hXO/(x»,

where hEIR ~ TslR, k E TxS and .i E .J(e, dale). The corresponding compatible Riemann metric is given by (1.10)

9J(t, x)((h 1, k 1), (h 2, k 2» = 0

0

(id x J)

=9j(7rO/k 1, 7rO/k2 ) + h1h2 + a(x)(kda(x)(k2) ' J -convex and its inside is contained in (-00,0) x S. The important feature of this class of almost complex structures is Fact 1.5([H)). Given a J-holomorphic disk u : D = {Izl ::; I} -t N contained in (-00,0) x S, if u(8D) C (-00,0) x S then u(D) C (-00,0) x S, i.e. u cannot touch S = {O} x S from the inside. Indeed, write u = (a, u) with a E IR and u : D -t S . From the proof of Lemma 19 in [H) we have It is easily checked that every hypersuface St = {t} x S is

~a = ~[J7r0/8sul~J + 17r0/8tul~J) 2:

°

on D .

But alaD < 0. Hence the strong maximum principle leads to the conclusion. Now using the diffeomorphism ~ in (1.8) we may define an almost complex structure on the open neighborhood ~(( -2€, 2€) X S) of S as follows: (1.11)

J w :=

d~ 0

J-

0

(d~)

-1

.

Guangcun Lu

294

It is clear that S is J<J?-convex.

But it is more important that every J<J?-

holomorphic curve contained in the interior of S cannot touch S from the inside. Notice that J<J? E .J(<J>(( -210,210) x S), w) . Denote by g<J? := wo (id x J<J?) the Riemann metric on <J>(( -210,210) X S) . Using the technique of a partition of unity it can be extended to a Riemann metric g on M with gl<J>([-f, 10] x S) = g<J? Then the standard technique gives rise to J E .J(M,w) satisfying: JI<J>([-f,f] X S) =

o

J<J? Clearly, this J satisfies the requirements.

2

Gromov-Witten invariants on compact symplectic manifolds with contact type bound. arIes

In [Lu] we defined the Gromov-Witten invariants on the compact semipositive symplectic manifolds with contact type boundaries as a special case of those on

a class of noncompact symplectic manifolds. In this section we will define them in general case. Let (M, w) be a compact symplectic manifold with smooth boundaries. Then the boundary 8M has a natural orientation defined by the volume form iv(wn)laM for an outward pointing vector field v along 8M . A contact form ,X on 8M is called positive if ,X 1\ (d,X)n-l = fiv(wn)laM for some smooth positive function f on 8M . Following A.Weinstein([WeJ) we call (M,w) to have contact type boundary if there exists a positive contact form ,X on 8M such that d,X = WlaM ' For such contact form ,X there always exist 10 > symplectic embedding of co dimension zero

°

and a

'ljJ: ((-10, 0] x 8M,d(e t ,X)) --t (M,w)

such that 'ljJ(O,x) = x for any x E 8M. Using it we get another compact symplectic manifold with contact type boundaries (M, w) as follows :

M

=M

U aM x{O}

8M x [0, 2]'

_ { w o n Mj w = d(et,X) on 8M x [0,2] .

Here t is the second coordinate. Let J), E .J(M,w) be the almost complex structure as constructed in Lemma 1.4. Then it it clear from construction of it that this J), may be extended onto all of M. We denote it by i)'. Notice that

Gromov-Witten invariants on compact symplectic manifolds

295

every hypersurface contact type {t} x 8M has the same property as 8M == {O} x 8M for any t E (-f, 2]). From now on we omit the subscript A in J>. without occuring of confusions. For any 0 < 8 < f we denote by (2.1) .J(M, 8M,w; 'I/J, J,8) := {J' E .J(M,w) I J'11/t((-o,Ojx8M) = JI1/t((-o,Ojx8M)}, { .J(M,8M,wj J) := {J' E .J(M,w) equals J near 8M}. Both are contractible nonempty subset of .J(M,w), and for any given JI E .J(M, 8M, Wj J) there exists a 8 > 0 such that JI E .J(M, 8M, Wj 'I/J, J,8). By perturbing J in .J(M, 8M, w; 'I/J, J,8) we may assume that .T is regular and fix it in the following arguments without special statements. Moreover, we are only satisfied pointing out how the arguments in [LT2] are realized in the present case. Let (1:: = U~l 1:: i j Xl, ... ,Xk) be a k-pointed semistable curve of genus g( which, by the definition, is connected[FO]), and 1I"i : ~i -t Ei denote the normalization of Ei . A continuous map u : E -t M is called J-holomorphic if every lift u 0 1I"i : ~i -t M is J-holomorphic. It is said to be stable if Ei contains at least three special points( double or marked points) when UiI::i is constant. Denote by [u, Ej Xl, ... ,Xk] the isomorphism class of the stable Jholomorphic map (u, 1::j Xl,"', Xk). Applying Lemma 1.4 to every component u 0 1I"i we"get u(E) n 'I/J(( -8,0] x 8M) = 0. This is a key for our arguments. Define u.([1::]) := Li(U01l"i).([~i]) the homology class represented by u. Given a A in the free part of H 2 (Mj Z) we denote by Mg ,k(M,w, J, A) the space of all isomorphism classes of k-pointed stable J-holomorphic curve of genus g. This is compact. Let Mg,k be the moduli space of Riemann surfaces of genus 9 with k marked points, and Mg,k be the Deligne-Mumford compactification of Mg,k consisting of the isomorphism class of all genus 9 stable curves with k marked points. Denote by --

EV: Mg,k(M,w,J,A) -t M the evaluation map given by EV([u, Ej Xl, '

"",

k

Xk]) = (u(xd,""' U(Xk)), and

the map given by successively contracting the unstable component of the domain of stable maps. Then, roughly speaking, the Gromov-Witten invariants

296

Guangcun Lu

are the homology class represented by the image of 7rg ,k x EV. To define them one need to embed Mg,k (M, w, J, A) into a larger space so as to get a "virtual fundamental class" of it. To this goal the notion of the stable CI-stable map( l ~ 1) with k marked points and of genus 9 was introduced in [LT2]. Let

-rA (M, g, k) be the space of all isomophism classes of such a map representing A. Then it contains Mg ,k(M,w,J,A) as a subset, and EV, 7rg,k are still well-defined on it. Notice that unlike the case of M being closed manifold the present -rA (M, g, k) has nonempty boundary. It is the property of the image of every map in Mg ,k(M,w , J,A) not intersecting with "p((-6, O] x 8M) that the space M 9 ,k (M, w, J, A) is contained in the interior of -rA (M, g, k) . That is, it does not intersect with the boundary part of it. Hence, by the compactness of Mg,k (M, W, g, k) we may choose an open neighborhood of it which is not intersecting with the boundary part of-rA (M, g, k) . So one only needs to repeat the arguments in [LT2] to define a generalized bundle E over -rA (M, g, k) and a natural section J given by the Cauchy-Riemann operator. They together give rise to a generalized Fredholm orbifold bundle with the natural orientation and of index r := 2c1(M)(A)+2k+(2n-6)(1-g). When l ~ 2 Theorem 1.2 in [LT2] determines an Euler class e([J : -rA(M,g,k) M ED in Hr(~(M,g,k);Q). But the inclusions FA(M,g,k) := F:(M,g,k) '-+ -rA(M,g,k) are all homotopically equivalient. Thus they give rise to a unique Euler class e([J : FA(M,g,k) homomorphism

M

ED

in Hr(FA(M,g , k);Q), which in turn defines a

by taking slant product of this Euler class by cohomological class in H*(Mg,k; Q). Clearly, it satisfies

P~',~,k(O U (3) = P~',~, k(O)/7r;,k(3 for any 0,(3 E H*(Mg ,k;Q) . Define

I)i('A~9,k) : H* (M; Q)k x H* (Mg,k; Q) -+ Q by 1)i('/9,k) ((3; 01, . .. ,Ok) = EV* (7ri 01 1\ ... 1\ 7rkOk)(P~:~,k ((3)). Notice that for any two almost complex structures J1 and J 2 in .:J(M,8M,w; J) there

Gromov-·W itten invariants on compact symplectic manifolds

297

exists a 8> 0 such that they belong to J(M, 8M, w; '1/;, J,8) and therefore are homotopic in J(M,8M,w;'I/;,J,8). We get that e([<JlJI : FA(M,g,k) I-t ED = e([<Jl h : FA(M,g,k) I-t ED and thus w,J 1

(2.2)

_

w,J2

PA,g,k - PA,g,k

and

"IJIw,h _ "IJI W,J2 (A,g ,k) - (A,g ,k) ·

These show that "IJI(A~~,k) is independent of the choice of J' E J(M, 8M, w; J) . Next, for the extended almost complex structure j = j;. on M above every .1' E J(M,w) may be extended into an element ], in J(M,8M,w; j;.)( but need not to be unique). Using the natural isomorphisms i* : H(M; Z) --t H*(Mj Z) and i* : H*(M; Z) --t H*(M; Z) induced by the inclusion i : M '-+ M we define

(13· ) .- ,T.-;;;J>. (13·, ell, - ... , elk - ) , ,k) , ell, ... , elk .- ~ (A,g,k)

,T.W,;' ~(A 9

(2.3)

for 13 E H*(Mg ,k; Q) and ell E H*(M; Q), l = 1, ···, k. Here A = i*(A) and III = (i*)-l(eld for any l. Clearly, this is well-defined. However, it may depend on the contact form A above. To show that such case cannot happen we denote by Cont+ (M, 8M; w)

(2.4)

the set of all positive contact forms A on 8M such that dA = WlaM. Lemma 2.1 The space Cont(M,8M;w) is a convex set. Proof For any two AO, Al E Cont+(M, 8Mj w), by definition it holds that dAO

=

WlaM = dAI, AO

1\

(dAo)n-1

= f(illwn)laM

and

Al

1\

(dAd n- 1 = g(illwn)laM

for some positive smooth functions f and 9 on 8M, Assume that (I-to)Ao(xo)+ toAI (xo)

= 0 for some to E (0,1) and Xo

E 8M. Then

But (1- to)f + tog> 0 on 8M. Hence the right side of (2.5) is not equal to zero at any point. This contradicts to the fact that (1 - to),o(xo) + toAI (xo) = o. Thus for any t E [0,1], I-form (1 - t)AO + tAl is nowhere zero. It is easily checked that they all belong to Cont+(M,8M;w).

0

Guangcun Lu

298

Define As := (1 - 8)AO + 8Al for 8 E [0,1]. Let ~s := KerA s and Ys be the Reeb vector field of A.•. A given Riemann metric g on 8M induces an Riemann metric gs on ~s. With gs and wle. the standard arguments produces a smooth family of almost complex structures is on ~s. On the other hand, Gray stable theorem gives rise to a family of diffeomorphisms Fs : 8M -+ 8M such that

F; As = fsAo for a family of smooth nowhere vanishing functions fs on 8M. Take a smooth family of vector fields X. om M such that

Define .)), t E [0,1] be exact. Then there exists an exact Lagrangian immersion F* : W' x S1 -+ SU X C such that the intersection Wo n W{ the set of double points of F*(W' x S1). Proof. See [5,2 . 3B~]. Now let c E C be a non-zero vector. We consider the equations

u=

(U1' U2) :

D -+ SU x C

(4.1)

= 0, 8U2 = c

(4.2)

UlaD: 8D -+ W

(4.3)

8U1

(4.4) Since W is a closed manifold in SU x C we know that

(4.5) here C1 depends only on W . Since every solution harmonic, it satisfies

U2

to the equation

8U2

= c is

Renyi Ma

312 and so there is no solution U = (Ul, U2) : (D2, 8D 2) --+ (SU equations for large Ilel!. i.e. we have proved

X

C, W) to this

Theorem 4.1 The Fredholm operator F : Vk(V, W,p) --+ E is not proper.

Proof. By Theorem 3.1 and 3.2, we know that the index of F is zero and deg(F) = 1, then the above argument show that F is not proper.

§5. The existences of holomorphic planes and periodic solutions In this section, we use the Sacks-Uhlenbeck-Gromov's trick to prove the existence of J-holomorphic disk with boundary in W if WE SU x C is Lagrangian submanifold. Now let e E C be a fixed non-zero vector satisfying Theorem 4.1. We consider the equations

(5.1)

U = (Ul' U2) : D --+ SU x C 8Ul

= O,8U2 = se

(5.2) (5.3) (5.4)

uI8D : 8D --+ W

In order to get the estimate of energy on solution u, we consider (V, W, d>.) = (SU x C, dN EB dx /\ dy) where SU is the symplectization of contact manifold (U,N). Now we recall the symplectization of (U,>.) is (SU,d(t>')) = ((Ux)O,oo[),d(t>.)) and put ~ = ker(>.). Then d>' is a symplectic structure for the vectorbundle ~ --+ U . We choose a complex structure J for ~ such that gJ := d>.o (Id x J) is a metric for ~ --+ U. As before we define an almost complex structure J on SU by

J(t, u)(h, k)

= (->,(u)(k), J(u)1fk

+ hX(u)),

(5.5)

where 1f : TU --+ ~ is the bundle projection along RX --+ U and X the Reeb vector field associated to N . We define a complete metric 9 on SU by

< (hI, kd, (h2' k2) >= hlh2 + >'(kd>'(k2) + gJ(1fk 1 , 1fk2)'

(5 .6)

By Lemma 3.1, we have a complete adapted Riemann metric on SU. So, we have an almost complex structure J = J 1 EB i on SU x C and adapted complete metric 9 = gl EB go on SU x C . Now for U = (Ul, U2) : (D2, 8D2) --+ (SU X C, W), define

r

8u8u 8u8u E(u) = JD(g(8x , J 8x)+g(8x , J 8x))dcr

(5 .7)

Exact Lagrangian submanifolds

Lemma 5.1 Let (V, W)

u

313

= (SU x C, W), J,g

as above and

= (Ul, U2) : D -t SU X C aUl = 0, aU2 = sc UI8D:

aD -t W

(5.8) (5.9) (5.10) (5.11)

Then, we have the following estimates

(5.12) here

C3

depends only on c, symplectic form w, W .

Proof. See (5) . Lemma 5.2 Let (W, V) = (SU x C, W), J, g as above and U = (Ul,U2) : D -t SU aUl

X

C

(5 .13)

= 0, aU2 = sc

(5.14)

aD -t W

(5.15)

UI8D:

(5.16) Then the image of U SUxC.

= (Ul, U2)

: (D2, aD2) -t (SU x C, W) is bounded in

Proof. That u2(D2) is bounded in C follows as above since W is closed and U2 is harmonic. Since Ul is J -holomorphic and Ul (aD) is bounded and its energy or area is bounded by the above Lemma, then the monotonicity of minimal surfaces concludes that Ul can not go to infinity. Theorem 5.1 There exists a non-constant solution U : (D2,aD2) -t (SU x C, W) of the partial differential equations Us

+ J(u)Ut = 0

(5.17)

where

(5.18)

Proof. By the above Lemma, we know that all arguments in [5] for the case V' is closed in (V' x C, W) can be extended to our case V' = SU is not closed since the images of maps remains in a bounded set.

314

Renyi Ma

Proof of Theorem 1.1. By the assumption of Theorem 1.1, we know that the Lagrangian submanifold W in SU x C is embedded and exact if Theorem1.1 does not hold. Then for large vector c E C we know that the nonlinear Fredholm operator or Cauchy-Riemann operator has no solution, this implies that the operator is non-proper. The non-properness of the operator implies the existence of J-holomorphic disk with boundary in W which contradicts the fact that W is exact since J -holomorphic disk has positive energy. For more detail, see [5].

References [1] Arnold, V.& Givental, A., Symplectic Geometry, in: Dynamical Systems IV, edited by V. I. Arnold and S. P. Novikov, Springer-Verlag, 1985. [2] Eliashberg, Y., New invariants of open symplectic and contact manifolds, Journal of AMS. 4(1991), 513-520. [3] Eliashberg,Y.& Gromov, M., Lagrangian Intersection Theory: FiniteDimensional Approach, Amer. Math. Soc. Transl. 186(1998) : 27-118. [4] Eliashberg,Y., Hofer,H., & Salamon,S., Lagrangian Intersections in contact gemetry, Geom. and FUnct. Anal., 5(1995): 244-269. [5] Gromov, M., Pseudoholomorphic Curves in Symplectic manifolds. Inv. Math. 82(1985), 307-347. [6] Hofer, H., Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjection in dimension three. Inventions Math., 114(1993), 515-563. [7] Klingenberg, K., Lectures on closed Geodesics, Grundlehren der Math. Wissenschaften, vol 230, Spinger-Verlag, 1978. [8] Ma, R., A remark on the Weinstein conjecture in M X R2n. Nonlinear Analysis and Microlocal Analysis, edited by K. C. Chang, Y. M. Huang and T . T. Li, World Scientific Publishing, 176-184. [9] Ma, R., Symplectic Capacity and Weinstein Conjecture in Certain Cotangent bundles and Stein manifolds. NoDEA.2(1995):341-356. [10] Sacks, J.& Uhlenbeck, K., The existence ofminimaI2-spheres. Ann. Math., 113(1980), 1-24. [11] Smale, S., An infinite dimensional version of Sard's theorem, Amer. J. Math. 87(1965): 861-866.

Low dimension anomalies and solvability in higher dimensions for some perturbed Pohozaev equation G. Mancini* Dipartimento di Matematica, Universita di Roma Tre Abstract In this talk, we will present a new solvability condition in higher dimension for a Brezis Nirenberg type equation. This result, which we believe to be sharp in some sense, has been obtained in collaboration with Adimurthi.

1

Introd uction

Very surprising "low dimension phenomena" have been observed by Brezis and Nirenberg in their pioneering work [BN] concerning perturbations of the "Pohozaev equation" -~u

=U

u>O u= 0

~ N - 2

+ p(u)

in 0

inO in 80

Here, 0 C !RN , N ~ 3 , is a smooth bounded domain and p is sub critical , with p' (0) = O. N Dealing with pes) = s1i1=2. Brezis and Nirenberg discovered an anomalous behaviour in dimension 3: (i)if N > 3, (P) has a solution '0

and N

k

2:: 5 , (P) has a solution.

If.f P(~) = 0 and f I\7HI2 > f p(~)W' then if N 2:: 7, (P) has a solution '

2N° 2N 4 Also, I IV, IN-2 = I ut -2 + ;::'2 I IU, + tw, Ii¥=> (U, + tw,)w, 2N N±2 I (1- t)IU +tw IN~2W2 = sIf +O(I)(f¥ + (~)N-2)~ + N-2 N-2 B~x[O,11 " , .r .

by Holder inequality, Lemma 3.2-ii and Lemma 4.1-i. Also, all estimates m (4.4) (4.5) are uniform in r 2 1. Now, from (4.4)-(4.5) and

we get

N)~

because I Jp(t,Y.)Y.1 ~ LNIY.I,.J~2(f2 2N

This proves the claim, because It, - 11 ~ N;-2I t

4

t- 2 -

11.

References [AM] Adimurthi and Mancini, A sharp condition in higher dimensions for a Brezis Nirenberg type equation, in preparation [AY] Adimurthi and Yadava, Critical Sobolev exponent problem in a ball with nonlinear perturbation changing sign, Advances in Differential equation 2,(1997), 161-182.

Some perturbed Pohozaev equation

325

[BG) Bemis F. and H. -Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators, J.Differ Equations 117, (1995), 469-486. [BN) Brezis H. and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical exponents, Comm.Pure. Appl. math.36 (1983), 437-477. [PS) Pucci P. and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures at Appl. 69 (1990), 55-83.

A NEW PROOF OF A THEOREM OF STROBEL Paul H. Rabinowitz· Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail: rabinowi 0 such that whenever v E r(~i-l' ~i)' 1:::; i:::; l, with I(v) = C(~i-l'~i ) and w is a solution of (HS) with Ilw - vllw1 .2(R,lRn) :::; v, then w = v. Then Strobel showed:

(1.4)

328

Paul H. Rabinowitz

Theorem 1.2 [1] If V satisfies (V1)-(V2) and (1.4) holds, then there are infinitely many actual solutions of (HS) heteroclinic from ~ to'TJ. The solutions are characterized by the amount of time they spend near the points ~i, 1 :=:; i :=:; £ - 1.

The arguments that Strobel employed to prove Theorem 1.2 were in the spirit of delicate variational deformation methods developed by Sere [5] and others, e.g. [6]-[7] to obtain so-called multibump solutions of Hamiltonian systems. More recently, Bertotti and Montecchiari [8] obtained an analogue of Theorem 1.1 when V depends on t in an almost periodic fashion and very recently Alessio, Bertotti, and Montecchiari [9] found an analogue of Theorem 1.2 for a class of slowly oscillating (in time) potentials. That the potential is slowly oscillating allows one to avoid the non degeneracy condition (1.4). See also [10] in this regard where a related result was obtained by different arguments. The goal of this note is twofold. First Strobel's result will be improved by weakening (1.4). Indeed the study of the nondegeneracy condition here is of independent interest. Secondly a new and very elementary constrained minimization argument will be used to prove the generalization. The new existence proof was motivated in part by recent work of Calanchi and Serra [11] and by [12] and [13]. The non degeneracy condition will be studied in §2 and in §3 the new existence proof will be carried out for the special case of a two chain where the main features are already present. The more general result and some extensions will appear in [13].

2. The All or Nothing Lemma In this section, the degree of degeneracy for minimizers of (1.1) will be studied in the simplest setting. Thus suppose that a, b E zn and Theorem 1.1 yields a solution, Q, of (HS) that is heteroclinic from a to b with I(Q) = c(a, b). Suppose further there are no heteroclinic £ chains Q1, ... , Q l from a to b with

e

L

I(Qi)

= c(a, b)

1

with £ > 1, i.e. we only have the simplest possible type of heteroclinic connection between a and b. For any k E Z, set TkQ(t) == Q(t-k) . Then TkQ E r(a. b) and by (Vd, I(TkQ) = I(Q). Thus all such time translates of Q are also minimizers of I in r( a, b) . Of interest here is the nature of the set of such minimizers of!.

A new proof of a theorem of Strobel

329

Let

S(a, b) = {q(O) E IRn I q E r(a, b) and I(q) = c(a, b)}. By the above remarks, q(O) E S(a, b) implies q(k) E S(a, b) for all k E Z . Therefore a, bE $(a, b). Let Ca(a, b) and Cb(a, b) denote respectively the components of $ containing a and b. Then somewhat surprisingly we have Lemma 2.3 (All or Nothing) : Either

(i) Ca

= Cb,

or

(ii) Ca = {a} and Cb = {b}. Remark 2.4 Observe that (i) occurs if V = V(q) since then I(TlJq) = I(q) for all () E IR and q E r(a, b). Alternative (ii) is surely true generically but it is difficult to give a concrete example of where it holds since the sets Ci , i = a, b, cannot be written down explicitly. However we note in passing that if the period, 1, of V in t is replaced by T, then for large T and V(t, q) =

°

such that (2.1)

for all q E r(a, b) such that I(q) = c(a, b) . Indeed if (2.1) holds, (HS) provides LOO bounds for ij and interpolation inequalities then imply uniform bounds in C 2 (IR, IRn) for such q. To verify (2.1), let (2.2)

Paul H. Rabinowitz

330

where Br(z ) = {x E IRn Ilx - zl < r} . If q E r(a,b) and q(t) E IRn\Bk(:~n) for t E [ 1 and 2:{ J(u p ) = C(~i-l'~i)' Let

Assume for 1 :S i :S

e

(*) S(~i-l'~i) nC~i_l(~i-I'~i) Hence by Lemma 2.3,

=¢

The main theorem can now be stated: Theorem 3.5 Assume (VI)-(V2) and (*) hold. Then (HS) has infinitely many solutions heteroclinic from ~ to 'T} and distinguished by the amount of time they spend near ~i' 1 :S i :S e- 1. Remark 3.6 Note that if (*) fails to hold, there is a continuum of heteroclinic solutions of (HS) joining ~i-l to ~i for some i. Thus (HS) always has infinitely many distinct heteroclinic solutions. Of course in the autonomous case, this family of solutions may just be the set of phase shifts of a single solution:

{reQ

I () E ~}.

Theorem 3.5 is a consequence of a more precise result. To formulate it, for < p« r « 1 = inf n Ix - YI· By (*) and Lemma 2.3, there are neighborhoods Oi I of ~i-l

X,Y E ~+, x « y means x is small compared to y. Let 0 x#vEZ

and

Oi ,2

of ~i' 1 :S i :S

'

esuch that j = 1,2.

Let m E ZU . Define

A new proof of a theorem of Strobel

333

Set

bm = inf I(q) .

(3.2)

qEX m

Note that it is only the difference in m i +l - mi, 1 :::; i :::; f -1, that is important in the definition of bm , i.e. if i* = (i, ... , i) E Z2l, (3.3) Now we have Theorem 3.7 For mj+l - mj sufficiently large, 1 :::; j :::; 2f - 1, there is a Qm E Xm such that I(Qm) = bm . Moreover Qm is a solution of (HS) heteroclinic from ~ to TJ with

j

IQm(t) - ~I :::; r,

t E (-oo,md,

IQm(t) - TJI :::; r,

t E

IQm(t) - ~il

[mu,oo],

(3.4)

:::; r,

The proof of Theorem 3.7 when f = 2 (and m E Z4) will be carried out. The general case involves the same ideas but is technically more complicated. For notational simplicity set 6 = (, 0 1 ,1 = O~, 0 1 ,2 = 0( , O2 ,1 = D(, and O 2 ,2 = D1/' The proof consists of several steps: (A) There exists a Qm E Xm such that I(Qm) (B) For any

€

> 0 and m2 -

= bm and

(3.4) is satisfied,

ml, m4 - m3 sufficiently large, bm :::; c(~, TJ)

+ €,

(C) An auxiliary problem which determines the choice of € , (D) The choice of m3

- m2,

(E) The completion of the proof. Proof of (A) : Let (qk) be a minimizing sequence for (3.2). Then the form of I implies (qk) is bounded in ~~~(]R, ]Rn) . Therefore as in §2, along a subsequence, qk converges weakly in Wl~~ and strongly in Lk;'c to Qm E E with Qm(±oo) E Qm otherwise satisfying the constraints of X m , and I(Qm) :::; bm . The first inequality in (3.4) will be verified next. Let

zn,

t:::;

ml -

1,

Paul H. Rabinowitz

334

Then

uniformly in large k . The cost as measured by

i:

L(qk)dt of qk going from

at t = -00 to 8B r (O for some b E (-00, md exceeds some "( = "((r) > O. Suppose qk does not satify

~

(3.6) Observing that 1(Pk) < 1(qk) via (3.5) and Pk satisfies (3.4) via (3.5). Thus qk could be replaced by Pk giving us a new minimizing sequence with (3.6) now being satisfied by Pk . An identical analysis proves that

(3.7) as p -+ 0 uniformly in large k and (3.8) (3.9) The L~c convergence of qk to Qm implies Qm satisfies (3.6) and (3.8)-(3.9). Moreover since r < ~, (3.6) and (3.9) imply Qm( -00) = ~ and Qm(oo) = 'f/. Hence Qm E X m .

Proof of (B) : It suffices to find q* E Xm such that

1(q*) ::;

c(~, 'f/)

+c.

Let Ql, Q2 be such that Ql E r(~,(), 1(Qd = c(~,(), Q2 E r((,'f/), and 1(Q2) = c((,'f/) . These functions are not unique. Indeed TjQi satisfies the same conditions as Qi for all j E Z and i = 1,2. Normalize Ql as follows : From {TjQl I j E Z}, choose Qi = Til Ql so that Qi(t) E C\ for all t ::; ml and Qi(t) rf. O{ for some t E (ml ,ml + IJ . Similarly from {TjQ 2 I j E Z}, choose Q2(t) so that Q2(t) E O( for t::; m3 and Q2(t) rf. O( for some t E (m3 ,m3 + IJ . For m2 = m2(p) sufficiently large, Qi(t) E O( for t ~ m2. Let

'P(t)

= (t -

m2)( + (m2

+ 1- t)Qr(m2) .

A new proof of a. theorem of Strobel

335

+ 2 and set

Let m3 > m2

1/J(t) = (t - (m3 - 1))Q;(m2)

for m4 = m4(p) sufficiently large, Q2(t) E by gluing to to (I::~~ for

Qil:: cpl::+1

+ (m3

01/ for t 2:

- t)(.

m4. Let q*(t) be defined

1/J1::_I to Q21:

3

•

Then as in (3.8),

I(q*):::; c(~,() +o(p) +o(p) +c((,1J):::; c(~,1J) +€

(3.10)

provided that p = p(€) is small enough. Step (C): Let

r*(~, () = {q E WI~'c2(1R, IRn) 1 q( -00) = ~, q(O) E 8(O£, U Oc;), q(oo) = (} and similarly r*((, 1J) = {q E WI~';(IR, IRn) 1 q( -00) = (, q(O) E 8(0( U 01/)' q(oo) = 1J} . Set c*(~, () =

inf

I(q)

inf

I(q).

qEr*(£,,()

(3.11)

and c*((,1J) =

qEr*«,1/)

Above arguments as in (A) show there exist functions PI E r*(~, () such that I(Pd = c*(~,() and P2 E r*((,1J) such that I(P2 ) = c*((,1J). Furthermore c*(~,() > c(~,() and c*(('1J) > c((,1J) . Indeed if e.g. c*(~,1J) = c(~,1J), then since PI E r(c () and [(PI) = c(~, (), PI is a solution of (HS) . Hence PI (0) E S(~,(). But PI (0) E 8(O£,UOc;) and by construction 8(O£,UOc;)nS(~,() = ¢. Consequently c* (~, 1J) > c( ~, 1J). Choose € so that O and TraceX'(u) < 0, for every u E JR 2 . In (23), it is assumed a stronger version of (H3). The argument employed here for the systems in more than two dimensions combines the direct method of Liapunov (21) with the method used by Olech (17). Given m ~ 3, we write JRm = JR2+n = JR2 X JR n , and we suppose that the plane JR2 is an invariant set on which the hypothesis of Theorem B is satisfied. Supposing the existence of a Liapunov function on JRm \ JR2 satisfying a Palais-Smale type condition with respect to the vector field X, we are able to show that the invariant two dimensional plane is a global attractor for the system (AS). Then, assuming two technical conditions, we verify that the origin is a global attractor for (AS). Since we suppose that the plane JR2 is an invariant set, we may write X = (L + H,G) with X((x,O)) = (L(x),O), for every x E JR 2 . Henceforth, we denote by Xm , m ~ 3, the family formed by the vector fields X = (L + H, G) defined on JRm which have the plane JR2 as an invariant set with L satisfying the hypothesis of Theorem C.

°

As observed above, our results for the systems in more than two dimensions are also based on the existence of a Liapunov function for the system (AS) . More specifically, we suppose (H5) There exists a function V E C l (JR 2+n , [0, 00)) satisfying (i) inf{V(u) : u

= (x,y), Ilyll ~ 6} > 0, for every 6 > 0,

(ii) (\7V(u),X(u)) < 0, for every u E JR2+n \ JR 2. It is worthwhile mentioning that condition (H5) does not imply that the origin is a global attractor for (AS). To be able to show that the plane JR2 is a

Global injectivity and asymptotic stability

343

global attractor for the system (AS), we introduce a version of the Palais-Smale condition for a functional with respect to a given vector field X:

Definition Given X E C (mm , mm), we say that V E C 1 (mm ,m) satisfies the (PS) condition with respect to X at level c E m, denoted (PS)(X,c), if every sequence (Uk) c mm such that V(Uk) -t c and (\7V(Uk), X(Uk)) -t 0, as k -t 00, possesses a convergent subsequence. Note that V E C 1 (mm, m) satisfies the Palais-Smale condition at level cE if it satisfies (PS)(,vv,c). Throughout this paper we let M denote the set x {O}. Observing that a solution of (AS) satisfies the (positive) semicomplete condition if it is defined on [0,00), in [23) it is proved the following basic result:

m m2

Lemma A ( The Fundamental Lemma) Suppose X E Xm , m ~ 3, satisfies (H5) with V satisfying (PS)(X,c) for every c > 0. Assume further the semicomplete condition for the solutions of (AS) . Then, the plane M is a global attractor for the system (AS). Lemma A was motivated by the observation that a version of the PalaisSmale condition may be combined with the direct method of Liapunov to study the behavior of the orbits of a dynamical system. Since the plane M is a global attractor for any semi-complete solution of the counter-example to the Markus-Yamabe Conjecture [4), we do not expect that the origin is a global attractor for (AS) under the hypothesis of Lemma A. To overcome such difficulty, we assume the following two technical conditions: (H6) There exist c, M, R > IIxil > R, Ilyll < p, we have

°

and p E (0,00) such that, for every u = (x, y),

(i) I(L(x).l ,H(u))1 ~ MV(u), (ii) (\7V(u),X(u))

~

-cV(u),

and (H7) There exists 8 E [0,1) such that lim IIH(x,y)II IIxll-+oo,lIyll-+o IIL(x) II

< 8.

-

In (H6), L.l represents the vector field orthogonal to L, obtained by a counterclockwise rotation. Now, we may state:

Theorem D Suppose X E Xm satisfies (H5)-(H7), with V satisfying (PS)(X,c) for every c > 0. Assume further the semi-complete condition for the solutions of (AS). Then, the origin is a global attractor for the system (AS) .

E. A. B. Silva and M. A. Teixeira

344

We should mention that Theorem D was strongly motivated by the counterexample to Markus-Yamabe Conjecture on m3. In our setting, the technical conditions (H6)-(H7) provide the necessary estimates for us to apply a version of Olech's argument, showing that the orbits converging to the plane should follow the flow of the orbits on the invariant plane. For the sake of completeness, we present the proofs of Theorem A and Lemma A in sections 2 and 4, respectively. In section 4, we also state two technical results that give us some estimates for the length of a curve on m2 which has the origin on a bounded component of its complement. Those estimates are used in the proof of Theorem D. In section 3, we present the proofs of Theorems Band C. In section 5, we give an outline of the proof of Theorem D. Finally, we reserve section 6 to present our final remarks on the the results considered in this article. There, we also recall a conjecture presented in [23) which is related to a recent result by Gutierrez-Teixeira [10).

2

Proof of Theorem A

In this section we present a proof of Theorem A [22). First. we recall a deformation lemma due to Chang [2), Proposition 2.1 (A Deformation Lemma) Suppose f E C1(E, m) satisfies (PS). Assume that a is the only possible critical value of f on the interval [a, b) and that a is an admissible level. Then, there exists a continuous map r : [0,1) x (fb \ Kb) -+ fb \ K b, so that (i) r(O, u) = u, VuE fb \ Kb (ii) r(t, u) = u, V (t, u) E [0,1) x (iii) f(r(l,u)) = a, VuE fb \ (Kb u r)

r

Before proving Theorem A, we also need to establish two preliminary results. Considering u, v E Sc(f), given in the hypothesis of Theorem A, we define Cl

= '"fEr, inf max f(r(t)), tE[O,lj

(2.1)

with

r 1 = hE C([O, 1), E) I ,(0) = u,

,(1)

= v}.

(2.2)

As a consequence of Proposition 2.1, we have

Lemma 2.2 Suppose c is an admissible level of f. Then. either Cl is a critical value of f, or there exists, E r 1 such that max f(r(t))

tE[O,lj

< c. -

>C

and Cl

(2.3)

Global injectivity and asymptotic stability

345

Proof: It is clear that c ~ Cl < 00. Moreover, when c < CI, Proposition 2.1 and a standard minimax argument imply that Cl is a critical value of f. Hence, we may suppose that Cl = c. Since c is an admissible level of f, there exists E > 0 such that K n f-l([C - E, C + ED = Kc . Furthermore, by (2 .1), we have 1'1 E r 1 such that max f(')'l(t)) < C+E.

tE[O,l)

-

Invoking Proposition 2.1 one more time, we obtain 7(t, u) satisfying (i)-(iii) with a = C and b = C + E. It is not difficult to show that 1'(t) = 7(1,1'1 (t)) belongs to r 1 and satisfies (2.3) . The lemma is proved. 0 We now define C2

= sup min f(')'(t)),

(2.4)

1'Er2 tE[O,l)

with

r 2 = bE C([O, 1], r) 11'(0) = u,

1'(1)

= v},

(2.5)

to obtain Lemma 2.3 Suppose that C is an admissible level of f and that r 2 =f 0. Then, either C2 < C and C2 is a critical value of f, or there exists l' E r 2 such that

f(')'(t)) = c, 'If t E [0,1].

(2.6)

Proof: We have that -00 < C2 ~ c. As in Lemma 2.2, C2 is a critical value of C2 < c. For C2 = C, we apply Proposition 2.1 for - f with a = -c , b = -c + E, and E > 0 sufficiently small. Then, arguing as in. the proof of Lemma 2.2, we find a path l' E r 2 satisfying (2.6) . Lemma 2.3 is proved. 0 Proof of Theorem A: Suppose that u and v are not in the same pathcomponent of Sc(f). Considering Cl and C2 given by (2.1) and (2.4), respectively, we claim that at least one of those numbers is a critical value of f. Effectively, if we suppose otherwise, Lemma 2.2 implies that r 2 =f 0. Consequently, by Lemma 2.3, we must have l' : [0,1] -+ Sc(f) such that 1'(0) = u and 1'(1) = v. But, that contradicts the fact that u and v are not in the same path-component of Sc(f). On the other hand, if f has no critical value d =f c, then, necessarily, Cl = C2 = C and ~ and v are in the same path-component of Sc(f). That concludes the proof of Theorem A. 0

f whenever

Remark 2.4 {i} It is clear from the proof that Theorem A holds if we assume (PS)c for C E [Cl' C2], where Cl and C2 are given by {2.1} and {2.4}, respectively. {ii} Replacing f by -fin {2.1}-{2.2} and {2.4}-{2.5}, we obtain two other possible critical values for the functional f.

E. A . B. Silva and M. A. Teixeira

346

We now state a natural generalization of Theorem A that was also proved in [22]. Given a topological space X, we denote by Hk(X) the k-th reduced singular homology with integer coefficients. Theorem 2.5 Suppose f E CI(E,JR) . Assume that c E JR is an admissible level of f and that H.(Sc(f)) is not trivial. Then f possesses a critical value d::fi c. Remark 2.6 In [22} it is proved that the Proposition 2.1 is true under a generalized version of the (PS) condition. Hence , we may conclude that Theorem A holds when that hypothesis is supposed.

3

Proofs of Theorems Band C

The following consequence of the level surface theorem provides the proofs of the Theorems Band C, Lemma 3.1 Suppose that X = (h,h) E C I (JR 2,JR2) satisfies X(O) = 0, Det(X'(O)) ::fi 0 and (H2)-(H3), with h satisfying (PS). Then, X satisfies (H8) X(u) ::fi 0, for every u E JR2 \ {O}, (H9) There exist P, a

> 0 such that

IIX(u)11

~

a, VuE

JR2,

Ilull

~ p.

Proof: Invoking the inverse function theorem, we obtain two open balls centered at the origin, BPi (0) C JR2, i = 1,2, such that X : BpI (0) --t JR2 is injective and B p2 (O) C X(Bpl (0)). Thus, to prove Lemma 3.1, it suffices to show that (H9) holds with P > PI and a = min{P2, c}, with c given by (H3). Arguing by contradiction, we suppose that there exists u E JR2 such that IIX(u)11 < a and Ilull ~ Pl · By our choice of a, we have that Ih(u)1 < c and X(u) E B(O,P2) . Furthermore, by (H2)and (H3) and we may suppose that CI = h(u) rf. h(K(X)) Now, let v E BpI (0) be such that X(v) = X(u). Since h satisfies (PS), by (H2) and Theorem A, there exists "( : [0,1] --t JR2 such that "(0) = v, "(I) = u, and hb(t))

= CI = h(u),

V t E [0,1].

(3.7)

Considering h : [0,1] --t JR2 defined by h(t) = hb(t)), for t E [0,1], we have that h(O) = h(l) = h(u) . Furthermore, from (H2), (3.7) and the implicit function theorem, we find to E (0,1) such that h'(to) =< hb(to)), "('(to) >= 0,

Global injectivity and asymptotic stability

347

< ft(-y(to»,')"(to) >= 0, and ,),'(to)::J. O. This implies that Det(X'(-y(to») = 0, contradicting our choice of Cl. The proof of Lemma 3.1 is concluded. 0 For the proof of Theorem B, given Sc(X) ::J. 0, we take Uo E Sc(X) \ K(X) and apply Lemma 3.1 to the vector field F(u) = X(u + uo) - c, obtaining that X(u) ::J. c for every u E m? \ {uo} . 0 Theorem C is a direct consequence of Lemma 3.1 and Olech's result:

Lemma 3.2 Suppose X E X 2 satisfies (H4), (H8) and (H9). Then, the system (AS) is globally asymptotically asymptotically stable.

Remark 3.3 (i) As observed in the introduction Lemma 3.2 is the key stone for the proof of the two dimensional Markus- Yamabe Conjecture. (ii) We also note that Theorems Band C are also true under the generalized version 0 the (PS) condition assumed in [23}.

4

Proof of Lemma A and Preliminary results

Given Uo E m2+n, we will show that Uo is attracted by M. By Theorem C and the fact that X E Xm , it suffices to consider Uo fj. with ')'(t) = ')'(t, uo) satisfying

m2

11')'(t)11 fi 0,

as t -+ 00.

(4.8)

Following the standard notation for Liapunov functions, we set V(t) { V(t)

= V(-y(t», = ~~ (t) = (\7V(-y(t»,X(-y(t»).

The next lemma is a consequence of (H5) and the fact that the origin is an asymptotic attractor for the system (AS) .

Lemma 4.1 Suppose X E Xm satisfies (H5). Assume,), = ')'(. ,uo) : [0,00) -+ (4 .8). Then, Ib(t)II -+ 00 as t -+ 00.

m2+n is a solution of (AS) satisfying

We now conclude the proof of Lemma A. As observed before, it suffices to consider,), = ')'(.,uo) satisfying (4.8). We claim that there exists a sequence tk -+ 00, as k -+ 00, such that

V(tk) -+ 0, as k -+ 00. Effectively, if we assume otherwise, we find T > 0 and K > 0 such that V(t) ::; -K, for every t ~ T. But this implies V(t) -+ -00 , as t -+ 00,

E. A. B. Silva and M. A . Teixeira

348

contradicting (H5). The claim is proved. Next, invoking Lemma 4.1 and the fact that V satisfies (PS)(X,c), for every c > 0, we conclude that V(tk) -+ 0, as k -+ 00. Therefore, V(t) -+ 0, as t -+ 00, since 0 < V(s) ::; V(t), for every s ~ t. By (H5), we finally get that Ily(t)11 -+ 0, as t -+ 00. 0 As a consequence of the proof of Lemma A, we have Corollary 4.2 Suppose X E Xm , m ~ 3, satisfies (H5). Assume 'Y(., uo) = (x(.),y(.)) : [0,00) -+ JR2+ n is a solution of (AS) satisfying (4.8). Then, Ilx(t)11 -+ 00 and Ily(t)1I -+ 0, as t -+ 00 . The next results provide estimates for the arclengths of curves on JR2 \ {O} that have the origin belonging to a bounded component of their complement. Given a continuous curve (3 : [0, 1] -+ JR2, we denote by 1((3) = 1((3([0, 1])) its length. Lemma 4.3 Suppose (3 : [0, 1] -+ JR2 is a continuous closed curve such that

((3d the origin belongs to a bounded component of JR2 \ (3([0, 1]), ((32) there exist to E [0,1] and d > 0 such that

11(3(to)11 ~ d > O. Then. 1((3)

~

2d.

Proof: Without loss of generality, we may suppose that to = o. By ((3d, there exist t E (0,1) and oX > 0 such that (3(t) = -oX(3(O). Consequently, by ((32), 1((3) = 1((3([0. t]) + l((3([t, 1]) ~ 2d. The lemma is proved. 0 Corollary 4.4 Let (3 : [0,1] -+ JR2 \ {O} be a piecewise C 1 simple closed curve satisfying ((32). Suppose T : JR2 -+ JR2 is a vector field of class Cl satisfying (Td T(O) = 0 and T(x) ::p 0, for every x E JR2 \ {O}, (T2) (T((3(t)), ((3'(t)).l) ~ 0(::; 0), for every t E [0,1] such that (3'(t) is defined. Then, 1((3)

~

2d.

Remark 4.5 Note that Lemma 3.1 implies that L, L.l : JR2 -+ JR2 satisfies (T1) whenever L satisfies the hypothesis of Theorem C.

Global injectivity and asymptotic stability

5

349

Proof of Theorem D

To prove Theorem D, we argue by contradiction and suppose that (AS) possesses a solution ')'(t) = (x(t),y(t)) = ')'(t,uo) satisfying (4.8) . As observed before, we have that ')'(t , uo) ¢ M , for every t 2:: O. In the following, we set F=L+H. Our main argument is a variation of Olech's method for the two dimensional problem. To use this argument, we consider x(t), the projection on M of the solution ')'(t,uo), and ')'(t,xo), the solution ')'(t , xo) of (AS) with the initial condition Xo E M. Then, Ilx(t)11 -+ 00 and Ih(t,xo)11 -+ 0, as t -+ 00, by Corollary 4.2 and Theorem C, respectively. Next, given T E m, we consider 1/(t,T) = 1/(t,')'(T,XO)), the solution in M of the system x(t) = L.L(x(t)), { x(O) = ')'(T,Xo) E M. Suppose the existence of T = T(T) and S = S(T) 2:: 0, T 2:: 0, such that x(S) = 1/(T, T) and T(T) -+ 00, as T -+ 00. If we know that

r = ')'([0, T)) U 1/«0, TJ, T) U x«O, S)) is a simple closed curve, we may apply Green's Theorem on the bounded component of m2 \ r to estimate the length of 1/([0, TJ, T). For this, we use an estimate for the function Ro(t)

= foo I(L(x(s)), F.L(x(s), y(s)))1 ds,

which we call the rate of the flow of Lacross x([O, 00)). That argument would provide a contradiction to Ilx(t)11 -+ 00 and ')'(T, xo) -+ 0, exactly as in Olech's argument [17] . However, we note that the claim that r is a simple closed curve is not true in our setting. Furthermore, we do not have T = T(T), S = S(T) well defined in general. To overcome these difficulties, given s, T > 0, we find sequences TO = 0 < T1 < .. . < Tj :::; T and So = 0 < Sl < ... < Sj :::; s such that, for every T E h-1,Tj), there exist unique T(T) and S(T) E [Sj-1,Sj) so that X(S(T)) = .,,(T(T), T) and l(1/([O, T(T)], T) < R. Moreover,

is a simple closed curve. Applying Green's Theorem on the bounded component of m2 \ r T, with T -+ Tj, after a finite number of steps, we verify that either Tj = T, or Sj = s ~ Then, taking T and s sufficiently large, we show that this contradicts the estimate for the length of 1/([0, T(T)J, T) .

E. A. B. Silva and M. A. Teixeira

350

Since the argument employed here is similar to the one used in [23], we just provide an outline of the proof of Theorem D, proving some of the main steps. Considering R, p'> 0 given by (H6), taking R > 0 larger and p > 0 smaller if necessary, we invoke the Lemma 3.1. X E Xm and (H7) to find d > 0 and o ~ 8 < 1 such that

IIL(x)11 ~ d > 0, { IIH(x,y)11 ~ 81IL(x)ll,

'if Ilxll ~ R, 'if Ilxll ~ R, Ilyll ~ p. Then, applying Corollary 4.2, we find T ~ 0 such that

IIx(s)11 ~ 3R, { lIy(s)11 ~ p,

'if s ~ T, 'if s ~ T. The following lemma provides an estimate for Ro(t) . Lemma 5.1 There exists Tl

Ro(T1 )

=

1

00

~ T, T

(5.9)

(5.10)

given by (5.10), such that

I{L(x(s)) , Fl.(x(s), y(s)))1 ds
0 such that f'(to) = 0. In the setting (3), f(t) satisfies f(O) = 0, f(oo) = -00 and there exists a unique to > 0 such that f'(to) = 0 and f(to) > O. Lastly in the setting of (2), we can see that f(t) is a constant function for every u. Since R x SN-l and RN \ {O} are diffeomorphic through a mapping

Kazunaga Tanaka

362

We can reduce our problem to the existence problem for closed geodesics on a non-compact Riemannian manifold R x SN~l with a metric gV defined by grs,z)

= e 2S (H -

(6)

V(eSx»grs,z) '

Here grs ,z) is the standard product metric on R x SN-1 :

grs,z)«~' 17), (~, 17» = 1~12 for (s,x) E R we identify

X

+ 11712

SN-1 and (~ , 17) E T(s ,z)(R x SN-1)

=R

x T z (SN-1). Here

= {17 ERN ; x ' 17 = O} . We remark that g(vs,x ) = gOts,x ) if V(q) = -r::b- and H = O. T z (SN-1)

Iql-

In what follows , we study the existence of periodic solutions under the situations which generalize (1)- (3) and the existecne of non-constant closed geodesics in a situation related to (2).

2

Strong force case

A situation related to (1) is called strong force and it is studied by [AmbrosettiCoti Zelati[l], Benci-Giannoni[8]' Greco[13]' Pisani[14]) . For a large class of V, [Pisani[14]) showed the existence of a periodic solution of (HS.1)-(HS.2) . The following result is a special case of Pisani's result. Theorem 1 (c.f. [Pisani[14)]) Suppose that V(q) satisfies (VO)-(V2) with Then for any H > 0, (HS.l)-(HS.2) has at least one periodic solution.

Q> 2.

Remark 1 Pisani's result is more general. He requires just V(q) E C 1(RN \ {O}, R) and his conditions admit more general behavior at 0 and 00 .

To prove Theorem 1, Pisani uses a minimax method based on the fact 1= 0: essentially he proves the following minimax value is a critical value of J(u) .

7rN- 2(A)

(7) where ~

= {a E C(SN-2,A); dega 1= O},

a(~)(r) .. SN- 2 x([0, l]/{O,I}) '" SN-2 x Sl --* SN-1 . a-(' O.

A perturbation problem for scale invariant functionals are studied for semilinear elliptic equations in RN by [Bahri-Li[4], Bahri-Lions[5JJ. Here we use an idea from [Bahri-Li[4]] . In addition to (VO)-(V2) we assume the following condition: (V3) Set W(q) = V(q)

+ dr-,

then W(q) satisfies

IqI2W(q), IqI3\7W(q), IqI4\7 2W(q) -+ 0 as Iql -+

00.

Prescribed energy problem for a singular Hamiltonian system

365

Our existence result is Theorem 4 ([Tanaka[19]]) Assume (VO)-(V2) with Q = 2 and (V3). Then (HS.l)-(HS.2) with H = 0 has at least one periodic solution. The conditions (V2) and (V3) require

V(q) '"

1

-jqj2

as

Iql '" 0 and Iql '" 00.

This condition is necessary for the existence of periodic solutions for (HS.1)(HS.2) with H = 0 in the following sense: if V(q) behaves like

V(q)

as

Iql '" 0,

(13)

V(q)

as

Iql '" 00

(14)

and a -:F b, then (HS.1)-(HS.2) with H = 0 does not have periodic solutions in general. In fact, if V(q) = - 2 it holds limu-+o IJJ~t (h2)

= b > O.

In h(X)(CPI(X))qdx < o.

Then there exists X> 0, X::; A*, such that if A2(0) < A < X and A f/. a( -.6.), {3.1} has at least seven nontrivial solutions. More precisely,

378

Zhi-Qiang Wang

(a) (3.1) has at least two positive solutions ui and ut where 4>(ut) < 0, ut is a local minimizer of 4>(u) . (b) (3.1) has at least two negative solutions u 3 and ui where 4>(ui) < 0, ui is a local minimizer of 4>(u). (c) (3.1) has at least three sign-changing solutions U5,U6 and U7 where U6 is a mountain pass point of4>(u), 4>(U6) < 0, ui < U6 < ut, ui < U7 < ut , and U7 is outside [ui, ut] .

Remark 3.17 If only>,! (n) < A < X and A (j. 0'( -~) are assumed, then (3.1) has at least five nontrivial solutions. More precisely, (a) and (b) still hold and there is a sign-changing solution U5 . This gives the nodal information of a result in [1].

Looking back at the proofs in the last section, we see our whole approach is based on the validity of Lemma 2.5. The negative gradient flow leaves the positive and negative cones invariant and ci(±P \ {O}) C ±P for t > O. However, in problem (3.1), the difficulty is that with the presence of hex) the nonlinear term is not order preserving any more, even under linear perturbations. In [30] we constructed a flow which may not be a pseudo gradient flow in general, but which still has some invariant property with respect to the positive and negative cones. This is done with an abstract framework and should be useful in other problems. We give a brief description of this construction. Let us make the following assumption.

(4)t) 4> E C 2 (E, R) and the gradient of 4> is of the form \74>(u) = Ao(u) -KE(U) where KE : E -+ E is compact, KE(X) C X and the restriction KE on X, K : X -+ X is of class C 1 and is strongly order preserving, i.e .. u > v => K(u) » K(v) for all u, v E X, where u »v ¢:> u - v E P; Ao : E -+ E is a locally Lipschitz operator such that Ail 1 : E -+ E exists, Ail 1 (X) c X and the restriction Ail 1 1x : X -+ X is of class Cl and strongly order preserving. Furthermore, there is M > 0 such that Ao(u) is linear for all u E X and Ilullx ::; M . Finally, we assume that there exists ao > 0 such that

and

Under the above assumption (4)1) ' if 4> also satisfies the (PS) condition, we shall construct a flow 1](t, u) along which 4> decreases and satisfies a deformation property, and under which ±P and ±P are both positively invariant, and finally

Sign-changing solutions for nonlinear elliptic problems

379

1l( t, u) leaves any order interval of the form [y., u) invariant provided Y., u is a pair of sub and super solutions with 1Iy.llx ~ M and Ilulix ~ M . Here we define Y. (u, resp.) a sub (super, resp.) solution if K(y.) > Ao(y.) (K(u) < Ao(u), resp.). Consider dtT~tt) = -Ail l (\7(O'(t,u))) {

0'(0, u)

= u.

Lemma 3.18

O'(t, ±P \ {O}) C ±P o 0 O'(t, ± P \ {O}) C ± P { O'(t, [y., u)) C [y., u) provided

1Iy.llx

~

M,

lIulix

~

'tit> 0,

"It> 0, "It> 0

M, and K(y.) > Ao(y.) and K(u) < Ao(u).

Proof. Let v E P \ {O} and consider v - Ail l (\7(V)). It suffices to show v - Aill(\7(V)) »0. This is equivalent to Ao(v) » \7(v) = Ao(v) - K(v) which follows from K(v) »0. Here we use the fact Ail l is strongly order preserving. Now, for any v E P such that y. + v ~ u,

follows from

which holds if

Ao(v) since K(y' + v) linear, we get

»

+ Ao(y.) > Ao(y. + v)

K(y') > Ao(y.). By the assumption, for

lIulix

~

M, Ao(u) is

o Remark 3.19 Note that -Ail l (\7(u)) is not a pseudo gradient vector field

for -\7(u), in general. We only get ±P are positively invariant. We do not know whether Uo ± P is positively invariant for any critical point Uo, like in Lemma 2.5. However, we still get the deformation property for using this flow. The following lemma is modeled on a deformation lemma in {43}.

Zhi-Qiang Wang

380

Lemma 3.20 Let SeE,

C

E R, £0

> 0, f> >

°such that

2£0 Vu E 4>-I([C - 2£0; C + 2£0]) n S2,s ~ 11V'4>(u)1I ~ ~. uao

Then 37] E C([O, 1] x E, E) such that (i) 7](t,u) = u, ift = or ifu r{. 4>-I([C - 2£O,c + 2£0]) n S20 . (ii) 7](1, 4>C+E n S) c 4>C-E. (iii) 7](t,·) is an homeomorphism of E, Vt E [0,1]. (iv) 117](t, u) - ull :::; f>, Vu E E, \:It E [0,1] . (v) 4>(7](t,u)) is nonincreasing, Vt E (0,1] . (vi) 4>(7](t, u)) < c, Vu E 4>c n S,s, Vt E (0,1]. (vii) 7] has the property of Lemma 2.1.

°

7](t, u) is built out of aCt, u), and for a proof of this lemma, see [30]. Now let us briefly describe how to apply this procedure to (3.1). Under (/5) and without hex), the usual method is to add a linear term on two side of the equation. Since h changes sign, this method fails. Our idea here is to add a nonlinear term on two sides of the equation. Let us define a function m(u) such that for lui:::; Mo for lui ~ Mo + 1 and m(u) is C 1 and monotonically increasing. Here, mo, Mo and p are constants which can be fixed according to feu) and hex) and the needs in the applications. Now consider

{

-/:::,.u + m(u) u

= AU + h(x)f(u)

+ m(u)

in n on an.

:= f(x, u)

=0

We choose Q < p < 2*, thus if mo and Mo are large, f(x, u) is strictly increasing. Let

4>(u)

=~

In

2

lV'ul dx

+

In

M(u)dx

-In

F(x, u)dx,

u E E,

where M(u) = Jou(x) m(s)ds . Then V'4> has a form

V'4>(u) where K(u)

= (_/:::,.)-1 f(x , u)

= Ao(u) -

K(u)

is strongly order preserving, and

(Ao(u) , v)

=

In

(V'uV'v

+ m(u)v)dx

Sign-changing solutions for nonlinear elliptic problems

381

i.e., Ao(u) = u + (-L:,)-lm(u) . Note that Ao(u) is linear for Ilullx ~ Mo. It is easy to see Ao E C l and Ao has a Cl inverse map Aill. By the above formula,

If u

= v, we get (u, Aol(u))

2: IIAill(u)112 .

The other requirements in (. In uh + In a(x)lulq-1uh + In f(x , u)h.

The inverse operator K of -A in the space E can be given by

(Ku , v)

= Inuv,

'r/u,v E E .

The operator K is selfadjoint and compact in space E. By the spectral theorem,

(Id - AK)u

= E~l (1 -

: )Pju, J

where Pj is the orthogonal projection of the space E on the eigenspace E j

.

A class of resonant or indefinite elliptic problems L e m m a 6 Suppose (a1) and (fl) hold wzth a(%) > 0 on the domazn Then, for any gzven A, there are posztzve constants p and 6 such that I(u)

> 6,

as u

E

E

and 11u//= p

T h e sketch of t h e proof for Lemma 6. As a(x) r 1, the lemma follows from the inequality

where fi > 1 is a fixed number and the constant c = c($) > 0. For general function a(+) > 0 without a positive lower bound, more machineries are needed to evaluate the functional. Suppose X = A, with i > 1. Let the subspace F, = El E2 . . . E, the oi.thogona1 sum of the subspaces. Thus E = F,-1 E, F .: Each element u of E can be decomposed as

+

+ +

+

u = UI

+

+UO

+uz,

where u~ € F , - ~ , u o E E,,uz E F :

It is easy to see that there is a positive constant [ such that ((Id - XiK)u, u)

< -[/lu/)"

((Id - X;K)u, u) = 0,

I

Vu E Fi-1, Vu € E;,

Vu E F:. ((Id - XiK)u,u) 2 [ I / U ~ / ~ , Namely, the quadratic part of the functional is only nonpositive on the finite dimensional subspace Fi. Then it can be controlled by the subquadratic part ' u = 0. The "small sphere" condition is I/UI)$+~ = A q t 1 w a ( z ) l ~ l ' ~ around maintained by the indirect arguments. For details, see [ll]. T h e proof for T h e o r e m 1. It remains to verify the PS condtion. I t can be done by the same idea as above. The boundedness of PS sequence on the : are obtained directly, while obtained by the equivalence spaces Fi-1 and F of the two norms I( . I/ and /I ./I, on the subspace F;. The proof for Corollary 2. It is easy to see that Eq. (D) has no nontrivial solution a t all for A 5 XI. As a consequence of Theorem 1, for each X > X I , there is always a solution uh. By the fact that the constant 6 in lemma 3 is independent of X as X is in a neighbourhood of XI, we can show that / l u ~/(, -t w as A, -t A.: Namely, there is a bifurcation from ( X I , m) for Eq. (D). As the function a(z) changes sign the above methods no longer work. First we give L e m m a 7 Suppose (al) and (fi) hold with p < q < 1. Then the functional I verifies the Palais-Smale condition a3 X # Xi, i E N. And the PS condition holds for X = X i if in addition (**) holds.

Shaoping Wu

388

The sketch of the proof for Lemma 7. It suffices to deal with the resonant case A = Ai . Suppose {un} in E is a PS sequence:

I(u n ) ::; d,

III (un)11 -+ o.

Decompose:

= Unl + UnO + Un2,

Un

where Unl E Fi-I , UnO E Ei,Un2 E Fl·

As in the proof for Lemma 6, there exists a constant 02 > 0 such that

II(Id -

AiK)UnW

>

(A~~l - I)211un1W + (A~:l - I)211un2W

> 02(llunll1 2+ IIUn211 2). So there is a constant 03 > 0 such that 0(1)

= III (un)11 2:: II(Id 2:: 03 (1lunlll + IIun211)

q cllunll P - c c.

AiK)Unll- cllunll - cllunll q - cllunll P -

It gives

IIunll1 2 ::; c(llunll1 2q + IIun211 2q + Ilunol1 2q ) + lower order terms of IIunl1 2q + 0(1) + c, since p < q. On the other hand,

~((Id -

I(un) =

AiK)un, un) + q ~ 1

In

q a(x)lunl +1 +

In

F(x, Un)

Ai ) II I2 1 Ai 2 > 21 ( 1- Al Unl I + 2(1- Ai+l )llun211

j.

- cllunll P+1- cllunll + -1- a(x)lunol q+1 q+1 n + ~I ( a(x)[lunlq+l -lunolq+1j.

q+ in Let A = In a(x)[lunlq+l - Iunolq+lj. By the inequality ab ::;

t + rk =

I, where

IAI
0 is an arbitrary constant, we have

f

In

IUno + B[Un2 + unlWlunl + un21

BE [0,1])

<
0 and c. -+ 0 as f --+ O. Therefore ,x(Ut)2 - g.(u.)ut

as u. > f,

InIV'UtI2 = In [,x(ut)2 -

(q

+ I)Gl on (Ut)

< ,x(ut)2 - G.(ut)

< c
O. It is still open whether the same is true assuming only that metric is smooth(see Yau [22]).

2.2

Nodal Set and Mores Index

In the article [19] I develop a type of Sturm-Liouville theory for certain superlinear elliptic PDE's as follows. Assume that 0 is a domain in JRN, and 6. is the standard Laplacian on O. For super-linear equations, under certain conditions it is known that there are infinitely many solutions with both Loo-norm and Morse index unbounded (see [1], also see [4] and references therein) . For general super-linear elliptic PDE's, after developing a new boundary estimate, I obtain the following estimate of the LOO norm of the solution via its Morse index:

IluIILOO(f1) :s G(1 + ind(u))"' , where ind( u) is the Morse index of the solution u. This result not only answers a question posed in Bahri-Lions [2], but also provides an explicit bound. Furthermore, using this result, as well as the Jerison and Kenig's Carleman-type inequality [14], and an iteration method from Donnelly-Fefferman [7], an estimate of the vanishing order r(x, u) ofthe solution is obtained of the following

Geometric structures of solutions

405

form: T(X, U) ::; 2C(1+ in d(u))"' .

Examples are known which indicate that one could not use only a purely local approach to prove the above estimate. Using my £,>0 estimate, as well as the Jerison and Kenig's Carleman-type inequality [14], and Hardt-Simon's estimates [12], we obtain an estimate of the N - 1 Hausdorff measure of the nodal set of the solution:

These are the first global estimates on the vanishing order and the measure for nonlinear partial differential equations. In this case, the left sides of these inequalities are the observable quantities, and the right sides are the functions of the Morse indices, which are defined for certain functionals over Sobolev spaces.

2.3

Singular Perturbations on Non-convex Variations

Consider the following singular perturbation problem Je(u)

= ~ In £/Vu/ 2 + ~W(U)dX, u E Hl(O) ,

where W E C(JR, JR+) has exactly two zeros a < (3, and 0 C JR2 is a bounded open region with smooth boundary. Je is the energy functional from the context of the Van der Waals-Cahn-Hilliard theory of phase transitions. Modica [17) studied the minimizers with the constraint on the density. It is natural to study the existence of non-minimal solutions and their geometric nature as a singular perturbation, which is a possible way to explore the multi-layer phenomenon. In a recent work [21), I started to consider the existence of non-minimal solutions and their geometric nature for the above functional. The functional has two unique global minimizers, u = a and (3. Since the functional may not be smooth and it is non-convex, so critical point theory of Ambrossetti and Rabinowitz [1) (also see [4)) cannot be applied here, but a minimax value of the mountain pass type can be defined as follows

CE = inf max JE(h(t)), hH099

Xue-Feng Yang

406

where r is the set of all continuous paths in Hl (0) joining Q and (3. "From the construction, for sufficiently small c:, d > 0, there are a path h£.6 E rand t£.6 E [0.1), such that

h£.6 is called to be an approximating mountain pass, and U£.6 = h£.6(t£.6) an approximating mountain pass point. Based on a fine analysis on the classical Dido's problem, I showed that the following result[21): If the domain is symmetric about origin, then for any set of approximating moutain passes, there exist approximating moutain pass points such that the interface of the limiie·~'l~~6blem contains finitely many arcs in 0 whose both -. 1 1 ~'" ends are in ao. ' It is worth to mention that if 0 is an ellipse, then the interface is its short diameter. Some nonsymmetric variants are also discussed in the paper [21].

2.4

Some Comments

The existence of the nodal point has not been discussed in this paper. Fortunately, some papers can been found in this proceeding, which are about the topic of the sign changing solution. This is one important way to approach this problem. The estimate of the lower bound of N-1 dimensional Hausdorff measure of the nodal set is still open. In the perturbation problem, we have a better understanding on the nodal set as discussed above. The result depends on the nonlinearity and the domain. There are some other examples which people can be found in this proceeding.

References [1] Ambrosetti, A. and Rabinowitz, P. H. Dual variational methods in critical point theory and applications. J. Funct. Anal., vol. 14, 369-381(1973). [2] Bahri, A. and Lions, P. L. Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math., vol. XLV, 1205-1215 (1992).

Geometric structures of solutions

407

[3] Benci, V. and Fortunato, D. A remark on the nodal regions of the solutions of some superlinear elliptic equations. Proc. Royal Soc. Edinburgh, lIlA, 123-128 (1989). [4] Chang, K. C. Infinite Dimensional Morse Theory and Multiple Solution Problrms. Birkhiiuser, PNDLE vol 6, (1991) [5] Coffman, C. V. Lyusternik-Schnirelman theory:complementary priciples and the Morse index. Nonlinear analysis, theory, methods and appl., vol.12, 506-529(1988). [6] Courant, R and Hilbert, D. Methoden der mathematischen physik. vol. 1, Springer, Berlin (1931). [7] Donnelly, H. and Fefferman, C. Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math., vol 93, 161-183 (1988) . [8] Donnelly, H. and Fefferman, C. Nodal sets of eigenfunctions: Riemannian manifolds with boundary. Analysis, Et Cetera, Academic Press, Boston, MA, 251-262 (1990). [9] Gel'fand, 1. M. and Levitan, B. M., On the determination of a differential equation from its spectrum, Izv. Akad. Nauk SSSR Ser. Mat., vol. 15, 309-360 (1951). [10] Gesztesy, F., and Simon, B. Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum. to appear in Trans. Amer. Math. Soc.(1999) [11] Hald, O.H. and McLaughlin,J.R, Solutions of inverse nodal problems. Inverse Problems, vol. 5, 307-347 (1989) [12] Hardt, R and Simon, L. Nodal sets for solutions of elliptic equations. J. Diff. Geom., vol. 30, 505-522(1989). [13] Hochstadt, H., and Lieberman, B. An inverse Sturm-Liouville problem with mixed given data. SIAM J. App. Math., 34, 676-680 (1978) [14] Jerison, D. and Kenig, C., Unique continuation and absence of positive eigenvalues for Schrodinger operators, Ann. of Math., 121, 463-494(1985). [15] Kreith, K. Nodal domain theorems for general elliptic equations. Rocky Mountain J . of Math ., vol. 1,419-425 (1971). [16] McLaughlin, J. R, Inverse spectral theory using nodal points as data-a unique result, J. DifJ. Equ., vol. 73, 354-362(1988).

408

Xue-Feng Yang

[17] Modica, L. Gradient theory of phase transitions and minimal interface, Arch. Rational Mech. Anal., vol. 98, No.2, 123-142(1987). [18] Yang, X. F . A solution of the inverse nodal problem. Inverse Problems, vol. 13, 203-213 (1997). [19] Yang, X. F. Nodal sets and Morse indices of solutions of super-linear elliptic PDE's. J. Functional Analysis, 160, 223-253(1999) . [20] Yang, X. F . A new inverse nodal problem. J. Differential Equations, a special edition dedicated to Hale's birthday(2000). [21] Yang, X. F., Non-minimal solutions of a singular perturbation on nonconvex variational problems: I, preprint (1999). [22] Yau, S.T. Problem section, Seminar on Differential Geometry. Princeton University Press, (1982)

KAM Theory for Lower Dimensional Tori of Nearly Integrable Hamiltonian Systems Jiangong You Department of Mathematics, Nanjing University Nanjing 210093, P.R.China E-mail: [email protected]

Abstract. In this article we give a brief survey for the recent developments in the KAM theory for lower dimensional tori of nearly integrable Hamiltonian systems and their applications in the study of invariant manifolds in the resonant zone and the construction of quasi-periodic solutions of Hamiltonian partial differential equations. The dynamics of the integrable Hamiltonian systems is simple in the sense that all the compact energy surface are foliated by invariant tori which carries quasiperiodic motions of the corresponding Hamiltonian equations. But integrable Hamiltonian systems are rather rare in the whole family of Hamiltonian systems. One of the landmarks in dynamical systems, especially in Hamiltonian dynamical systems, is the KAM (Kolmogorov-Arnold-Moser) theory, which tells us that for all Hamiltonian systems in a open neighborhood of a nondegenerate integrable Hamiltonian systems, the quasi-periodic motions in invariant tori are typical (See [Arnold, Encyclopedia] and the references therein). Later [Melnikov, 1965] formulated a KAM type persistence result for lower dimensional tori of nearly integrable Hamiltonian systems. In recent years, the KAM theorem for lower dimensional tori attracts more attentions for its applications in the construction of quasi-periodic solutions of Hamiltonian partial differential equations and the study of the dynamics in the resonant zone of nearly integrable systems.

1

Persistence of lower dimensional tori

We start with a real analytic Hamiltonian

H(x, y, u) = h(y, u)

409

+ P(x, y, u)

(1.1)

Jiangong You

410

in (x, y, u) E Tn X D X R 2m C Tn X Rn X R 2m with the symplectic structure 2:~=1 dXi /\ dYi + 2::'1 dUi /\ du_;. The corresponding Hamiltonian equations are dy - = -Px, dt where J is the standard symplectic matrix in R2m. Assume that hu(Y,O) = O,dethuu(Y,O)"# 0, and P is small, i.e., the unperturbed Hamiltonian system defined by h possesses an invariant sub-manifold x E Tn, y ED, u = 0 foliated by a family of invariant tori x E Tn, y = Yo, u = 0 and the flow on each torus is given by x(t) = Xo + hy(Yo, O)t, Y = Yo, u = O. If P is not zero, the sub-manifold x E Tn , y E D, u = 0 is no longer invariant. By KAM theory, one could expect that a Cantorian submanifold with positive Lebesgue measure in x E Tn, y ED, u = 0 persist the perturbation. Expanding the Hamiltonian (1.1) in the neighborhood of u = 0, we have

H(x, y, u)

= h(y, 0) + ~(huu(Y, O)u, u) + P(x, y, u) + O(u 3).

For the unperturbed system, the local normal behavior of the invariant torus y = Yo, u = 0 is determined by the matrix huu(Yo, 0) if it is non-degenerate. Linearizing the Hamiltonian in the neighborhood of torus Tn X {y = ~ E D} X {u = O}, we arrive at a family of perturbed integrable Hamiltonians,

H

1 = N + P = (w(~), y) + 2(A(~)u, u) + P,

(1.2)

where (x,y,u) E Tn X Rn X R 2m, w(~) = hy(~,O),A(~) = huu(~,O), and P = P + 02(y) + O(yu) + 03(U). We shall treat ~ E D as an independent parameter. This reduction reduces the persistence problem of invariant tori of a fixed Hamiltonian system into the persistence problem of an invariant torus of a family of perturbed linear Hamiltonian systems. The setting has been frequently used by many authors. In this paper, instead of (1.1), we shall formulate all results for (1.2) with an independent parameter ~ varying over a positive measure set D. For simplicity, we assume that H is analytic in all variables including ~ . If the unperturbed torus is elliptic, i.e., all the eigenvalues of JA(~) are pure imaginary and simple, (1.2) can be transformed to the following Hamiltonian n

H

1

m

= N + P = LWj(~)Yj + 2 L j=1

j=1

nj(~)(u; + U~j) + P,

(1.3)

KAM theory for lower dimensional tori

411

by a linear symplectic coordinator transformation. Melnikov proposed the following non-resonant conditions to guarantee the persistence of lower dimensional tori: (k,w(~))

+ ni(~)

:i 0,

(1.4)

+ ni(~) + nj(~) :i 0, (k,w(~)) + ni(~) - nj(~) :i 0, Ikl + Ii - jl ::f: 0, (k,w(~))

(1.5) (1.6)

where w = (WI,'" ,wn ). For the persistence result, we only need to consider (1.2) or (1.3) in the complex domain D(r,s)

= {(x,y,u)1

IIImxl 0 and b ~ 1 such that dn ::; d for all n , and (3.6)

KAM theory for lower dimensional tori

419

where An are real and independent of ~ while En may depend on ~j furthermore, the behavior of An 'S is assumed to be as follows Am - An -0 b b = 1 + o(n ), m -n

(A2) Gap condition: There exists

()l

n < m.

(3.7)

> 0 such that

(0'(-) denotes "spectrum of ''' ). Note that, for large i, j, the gap condition follows from the asymptotic property. (A3) Smooth dependence on parameters: All entries of En are smooth functions of ~ . (A4) Non-resonance condition: meas{~ EO:

(k,w(~))«k,w(~))

+ A)«k,w(~)) + A+ JL) = O} = 0,

(3.8)

for each 0 '" k E Zd and for any A, JL E Un EN a(On) j meas == Lebesgue measure. (AS) Regularity of the perturbation: The perturbation P is regular and small in some weighted norm sense (See P6schel[23j for details) . Roughly speaking, regularity means Xp maps a sequence with decay to a sequence with faster decay. Now we can state the KAM Theorem. Theorem 14 ( Cherchia and You[8], 1999) Assume that N satisfies (A1) (A4) and P is regular and small in the sense of (AS) and let 'Y > O. There exists a positive constant € = €( d, d, b, (), ()l , a- a, L, 'Y) such that if IIXp II < € , then the following holds true. There exists a Cantor set 0"'( C 0 with meas( 0 \ O"'() -+ 0 as 'Y -+ 0, such that the Hamiltonian equations governed by H = N + P on the Cantor set 0"'( has d-dimensional H -invariant torus.

The above Theorem has been obtained by P6schel[23j for d = 1, which applies to some ID PDEs with Dirichlet boundary conditions. The main idea has appeared earlier in, i.g., Kuksin[17]' Wayne[26j . Theorem 14 can be applied to the construction of the quasi-periodic solutions for ID wave equations with periodic boundary conditions.

Jiangong You

420

Theorem 15 (Cherchia and You[8], 1999) Consider a family of lD nonlinear wave equation {3.1} subject to periodic boundary conditions, parameterized by ~ == W E 0 ( ~ is chosen to be a fixed family of the eigenvalues) with V (-, ~) real-analytic {respectively, smooth}. Then for any 0 < 'Y « 1, there is a subset 0"( of 0 with meas(O\O"() -+ 0 as'Y -+ 0, such that (3.1)~Eo'Y has a family of small-amplitude {proportional to some power of 'Y}, analytic {respectively, smooth} quasi-periodic solutions of the form

n

where

Un:

r

d

-+ R and w~, ,·· ,w~ are close to

WI, '"

,Wd.

The existence result for (3.1) with Dirichlet boundary condition, obtained by Wayne and Kuksin, has been well known .

4

Bourgain-Craig-Wayne method

Before You[31] and Chierchia and You[8], people generally believed that the KAM theory was not compatible with multiple normal frequencies and thus could not be applied to ID Hamiltonian PDEs with periodic boundary conditions and higher dimensional case where the multiplicity of normal frequencies is essential. [Craig and Wayne, 1993] introduced a new approach to overcome the difficulties caused the multiplicity. They succeeded in proving the existence of periodic solutions of ID wave equations with the periodic boundary conditions. By improving in an essential way Craig and Wayne's technique, Bourgain proved the exisitence of quasi-periodic solutions for perturbed ID nonlinear Schrodinger [Bourgain, 1994]' ID wave equations and, most notably, 2D Schrodinger equations [Bourgain, 1998]. Their approach is based on a Liaponov.. Schmidt decomposition, which involves a multiscale analysis for controlling a Green's function. In this small survey, we shall not try to give a detailed exposition for this significant new approach since it is quite different from the standard KAM approach. In the following, we only state a recent result obtained by [Bourgain, 1998] to show the power of Bourgain-Craig-Wayne method. For details , we refere to [Craig and Wayne,1993], [Craig, 1996] and [Bourgain, 1998]. Consider the Schrodinger equation

.au at -

I

!':!.u + Muu

au

+ f au'

u

= u(x, t), x E T2,

(4.1)

KAM theory for lower dimensional tori

421

where H = E ajlUjUI is real analytic near 0; MCT is a Fourier multiple with MCTei(n,x) = (Tjei(n,x) if n = nj, j = 1,2" .. ,d, MCTei(n,x) = 0 otherwise. Take (T as parameters which varies over a bounded set 0 with positive measure. Theorem 16 [Bourgain, 1998] Let uo(x, t) = E:=l aje i ((n j ,x)+l' j t) , where J.t = Injl + (Tj and aj E R \ {o} . If E lajland 10 are small enough, then there exists a set 0(10, a) C 0, with limHo measure 0 \ 0(10, a) = 0, such that for any (T E 0(10, a), equation (4.1) has a quasi-periodic solution u(x, t), with frequencies Aj '" J.tj, which is close to Uo .

Final remark. So far, whether Theorem 16 has higher dimensional extention is still open. Also there is no counterpart of the result for 2D nonlinear wave equations. The reason has been explained in [Bourgain, 1998]. We hope that the combination of the KAM method and the Bourgain-Craig-Wayne method would be helpful for the problems.

References [1) Arnold, V. I. , Dynamical Systems III, Encyclopaedia of Mathematical Science, SringerVerlag, 1985. [2) J .Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, International Mathematics Research Notices, 1994, 475-497. [3) Bourgain, J ., Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schriidinger equations, Annals 0/ Mathematics,148(1998) , 363-439. [4) Broer, Hendrik W . Huitema, George B. Sevryuk, Mikhail B. Quasi-periodic motions in families of dynamical systems: order amidst chaos, LNM 1456, 1996. [5) Cheng, C.Q, Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltoniao systems Communication in Math. Phys. 171,1996,529-559. [6) Cheng C.Q., Lower dimensional invariant tori in the region of instability for nearly integrable Hamiltonian systems, Commun. Math .Phys., 203(1999), 385-419. [7) Chierchia L., and Gallavotti G., Drift and diffusion in phase space, Ann. Inst. H. Poincare Phy. Th ., 69(1994), 1-144. [8) Chierchia L. and You J., KAM Tori for 1D Nonlinear wave equations with periodic boundary conditions, To appear in Communication in Mathematical Physics, 2000 . [9) Cong F ., Kiipper T ., Li Y., You J., KAM-Type theorem on resonant surfaces for nearly integrable Hamiltoniao systems, To appear in J. Nonlinear Science, 1999. [10) Craig, W., KAM theory in infinite dimensions, Lectures in Applied Mathematics, 31, 1996. [11) Craig, W., and Wayne, C.K, Newton's method and periodic solutions of nonlinear wave equations, Commun. Pure Appl. Math ., 46, 1993, 1409 -1498.

422

Jiangong You

[12] Graff, S.M., On the continuation of stable invariant tori for Hamiltonian systems,J. Differential Equations, 15, 1974, 1-69. [13] Eliasson L.H., Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sc. Norm . Sup. Psia 15, 1988, 115-147. [14] Eliasson L .H., Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Mat. 25(1994), 57-76 [15] Kuksin , S.B., Nearly integrable infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Springer, Berlin, 1556, 1993. [16] Kuksin, S.B ., Poschel, J., Invariant cantor manifolds of quasiperiodic oscillations for a nonlinear Schrodinger equation, Annals of Mathematics, 142,1995149-179. [17] Kuksin S.B., Perturbation theory for quasi-periodic solutions of infinite-dimensional Hamiltonian systems, and its applications to the Korteweg-de Vries equation, Matern. Sbornik 136(118):31988, English transl. in Math. USSR Sbornik 64 (1994) , 397-413. [18] Lancaster P ., Theory of Matrices, Academic Press LTD, New York and London, 1969. [19] Melnikov V.K., On some cases of the conservation of conditionally periodic motions under a small change of the Hamiltonian function, Soviet Mathematics Doklady, 6, 1965, 1592-1596. [20] Moser J., Convergent series expansions for quasiperiodic motions, Math. Ann., 169(1), 1967, 136-176. [21] poschel,J ., Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici, 11, 1996, 269- 296. [22] Poschel J., On elliptic lower dimensional tori in Hamiltonian systems, Math . Z., 202, 1989, 559- 608. [23] Poschel J., A KAM-Theorem for some Nonlinear Partial Differential Equations, Ann. Scuola Norm. Sup. Pisa CI. Sci., 23, 1996, 119-148. [24] Rudnev M . and Wiggins S., KAM theory near multiplicity one resonant surfaces in perturbations of A-priori stable Hamiltonian systems, J. Nonlinear Science 1(1997), 177-209. [25] Treshchev D .V., Mechanism for destroying resonance tori of Hamiltonian systems, Mat . USSR. Sb. 180(1989), 1325-1346 [26] Wayne , C.E., Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory, Comm. Math . Phys., 121, 1990, 479-528 . [27] Xu J ., Persistence of elliptic lower dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems, J. Math. Anal. and Appl. , 208(1997), 372-387. [28] Xu J . and You J ., A symplectic map and its application to persistence of lower dimensional invariant tori for nearly integrable Hamiltonian systems, Preprint, 1999. [29] Xu J. and You J., Persistence of lower dimensional tori under the first Melnikov's conditions, Preprint, 1999. [30] You J., A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Commun. Math. Phys., 192( 1998), 145-168. [31] You J., Perturbation of lower dimensional tori for Hamiltonian systems, J. Differential Equations,152( 1999), 1-29. [32] You J ., Lower dimensional tori of reversible Hamiltonian systems in the resonant zone, To appear in Proceedings of the Conference in dynamical systems in Memory of Prof. Liao Shantao, World Scientific, 1998.

Periodic Solutions for N-Body Problems* Shiqing Zhangt Department of Applied Mathematics, Chongqing University Chongqing 400044, People's Republic of China Dedicated to the 60th birthday of Professor Paul Rabinowitz

Abstract For Newtonian N-body problems with any positive masses we prove the isolated properties of the collisions for the generalized solution corresponding to the critical point of the Lagrangian action, and we also prove that the minimizer of the Lagrangian action defined on the periodic orbit space on which each body has equal integral mean has no binary and triple collisions neither other collisions under some assumptions.

1

Introduction and Main Results.

N-body problems are related to solving the Newtonian equations:

.. miqi

aU(q)

= -a--' qi

i = 1, ... ,N,

(1.1)

where mi (i = 1, ... , N) is the mass of the i-th body and qi E Rk is the position vector of the i-th body, , q = (ql, .. . , qN), U(q) is the Newtonian potential (1.2) References [2], [4]-[8], [10], [13], [14], [18], [21] used the variational methods to study the solutions of (1.1)-(1.2) . In particular, for the case k = 3, Dell'Antonio ([8]) proved that the minimizer of the Lagrangian action defined on anti-T /2 periodic functional space has no isolated simultaneous collision under some assumptions on the masses of N -bodies, and pointed out some ideas for proving that the minimizer has no isolated two- or three-body collision. • Partially supported by the NSF of China. tThe author would like to thank Professors Long Yiming and Paul Rabinowitz .

423

Shiqing Zhang

424

In this paper, we firstly prove that all kinds of collisions for the critical point of the Lagrangian action are weakly isolated and then generalize the result of Dell'Antonio ([8]). Definition 1.1 ([4], [2]) Given r > 0, let qi E W 1 ,2([0, r], Rk), we call q = (ql, ... , qN) is a generalized solution of (1.1 )-(1.2), if there hold: (i) S(q) = {t E [0, r] I :11 :::; i i= j :::; N, s.t., qi(t) = qj(t)} has zero Lebesgue measure. (ii) For all t E [0, r] \ S(q), q(t) satisfies (1.1). (iii) For all t E [0, r] \ S(q), there holds

Definition 1.2 A collision time to E [0, r] is called weakly isolated or isolated if there is a neighbourhood of to such that there is no the same kind of collisions or no collisions in the neighbourhood of to. Theorem 1.1 The collisions of the generalized solution for (1.1)-(1.2) are weakly isolated. Given T > 0, let

(1.3)

1

r L mdcii(tWdt + inr U(q)dt T

f(q) = 2" in

o

T

N

i=l

(1.4)

0

Definition 1.3 ([8]) we call (1.1)-(1.2) satisfy the condition (Hjd: For the givep masses ml,· .. , mK, every nonplanar central configuration ([1], [3], [15]. [19], [20]) for K bodies is the isolated modulo rotation. Theorem 1.2 The minimizer q(t) of f(q) on A has no binary and triple collisions; and if condition (H K) holds, then the minimizer has no collision for K-bodies.

2

The Proof of Theorem 1.1.

Lemma 2.1 The binary collisions of the generalized solution for Newtonian N -body problems are weakly isolated.

Periodic solutions for N-body problems

425

Proof Without loss of generality, we assume t = 0 is a binary collision time for m1 and m2. We claim that t = 0 is a weakly isolated collision time. Otherwise, there is a sequence tn -+ 0 such that d12(t) = Iq1(t) - q2(t)1 satisfies d12 (t n ) = O. Hence d12 (t) attains the maximum value at some in E (tn, tn+d, so we have 2 (2.1) d d12 (t) t=tn ::; 0 dt2

I

Let ~ = ~(t) =

(2.2)

q1(t) - q2(t)

Then (1~12)' = 2~ . ~

:2 (1~12) =

(2.3)

21~12 + 2~ . ~

(2.4)

By the first two equations of the system (1.1) we know that (2.5)

Hence

~(1~12) = 21~12 _ 2(m1 + m2) dt 2 Iq1 - q21 In the following, we will further prove that ~1~12 _ m1 + m2 2

d 12

+ 0(1),

= 0(1)

(2.6)

(2.7)

We use the energy formula N

h =

~L

m;!qil 2

-

U(q)

i=1

(2.8)

Shiqing Zhang

426

Summing the first two equations of the system (1.1), we have

(2.9) By the last N - 2 equations of (1.1), we have

miqi(t)

E

C 2, as t

-t

0

(2.10)

By (2.8)-(2.10) we have (2.7) . Hence by (2.6) and (2.7) we have

~( leI 2) = 2(ml + m2) + 0(1)

dt 2

d 12

(2 .11)

Then (2.11) contradicts with (2.1) since from (2.11) we have

d2

dt 2 (leI 2 )

-t +00

as t

-t

0

(2.12)

Lemma 2.2 The triple collisions of the generalized solution of (1.1)-(1.2) are weakly isolated. Proof Without loss of generality, we assume t = 0 is a triple collision time for ml, m2 and m3. If t = 0 is not weakly isolated, then there is a sequence tn -t 0 of the triple collision times such that

423(t)

=

L

m,mjlq,(t) - qj(t)12

(2.13)

1~i<j9

=

=

satisfies d'f23(tn) d'f23(t n+1 ) 0, hence d'f23(t) attains the maximum value at some tn E (tn, tn+d, so we have (2.14) For 1

~

i

<j

~

3, let (2.15)

Then

We notice that

Periodic solutions for N-body problems

427

Where (2.17)

We notice that

Shiqing Zhang

428

(2.20) We use the energy representation: h

=

= (2.21) Summing the first three equations of the system (1.1), we know that 3

Lmiqi(t) i=l

E

C 2, as t -+ 0

(2.22)

By the i-th (4 ::; i ::; N) equation, we know that

miqi(t) E C 2, as t -+ 0

(2.23)

By (2.20)-(2.23) we have

~

m L m ·m 'I{ .12 + M3 (-m 1 m 2 - mlm3 - m2 3 ) - 0(1) (2.24) 2 0 such that for 1 ~ i "I l ~ n,

(3.23)

We set I(i - (d

= Cil > 0

(3.24)

Then for 0 ~ r ~ 2coc-3/2, we have

r 4/ 3 2C2

sC~

+ -2-

< Ir2/3(~i - ~l) + SW((i - (IW < 2C2r 4 / 3 + 2sclL

(3.25)

Then

Notice that for 2coc- 3/ 2 -+ +00, the integral: 1

1

ds

12000-3/2

C2 S il dr 12C2r 4/3 2sClL 13 / 2

+

(3.27)

converges absolutely to a positive constant. Hence there is C3 > 0 such that (3.28)

Shiqing Zhang

440 We notice that for 2eOe- 3 / 2 ~

+00,

(3.29) We notice that

(3.30) Where (3.31)

= ( (2e

io

o

+

(T) iT-2eo

L

[mim j _ mimj ] dt l