Critical Point Theory and Its Applications

CRITICAL POINT THEORY AND ITS APPLICATIONS

CRITICAL POINT THEORY AND ITS APPLICATIONS

By WENMING ZOU Tsinghua University, Beijing, China MARTIN SCHECHTER University of California, Irvine, California, USA

^

Spri ringer

Library of Congress Control Number: 2006921852 ISBN-10: 0-387-32965-X

e-ISBN: 0-387-32968-4

ISBN-13: 978-0-387-32965-9

Printed on acid-free paper.

AMS Subject Classifications: 35J50, 58E05, 47J30, 49505, 58E30 © 2006 Springer Sciencen-Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 springer.com

W. Zou dedicates this book to his parents: LIANG-SHENG ZOU & GUO-XIU ZENG. M. Schechter dedicates this book to his wife, children and grandchildren (currently 22) and great grandchildren (currently one).

Contents Preface

ix

1

Preliminaries 1.1 Partition of Unity in Metric Spaces 1.2 Sobolev Spaces 1.3 Differentiable Functionals 1.4 Topological Degrees 1.5 An ODE in Banach Space 1.6 The (PS) Conditions 1.7 Weak Solutions

1 1 2 7 13 16 19 20

2

Functionals Bounded Below 2.1 Pseudo-Gradients 2.2 Bounded Minimizing Sequences 2.3 An Application

25 25 26 31

3

Even Functionals 3.1 Abstract Theorems 3.2 High Energy Solutions 3.3 Smah Energy Solutions

37 37 44 49

4

Linking and Homoclinic Type Solutions 4.1 A Weak Linking Theorem 4.2 Homoclinic Orbits of Hamiltonian Systems 4.3 Asymptotically Linear Schrodinger Equations 4.4 Schrodinger Equations with 0 G Spectrum 4.5 The Case of Critical Sobolev Exponents 4.6 Schrodinger Systems 4.6.1 The Superlinear Case 4.6.2 The Asymptotically Linear Case

55 55 64 73 74 89 101 102 112

viii

CONTENTS

5

Double Linking Theorems 5.1 A Double Linking 5.2 Twin Critical Points 5.3 Eigenvalue Problems 5.4 Jumping Nonlinearities

117 117 120 129 131

6

Superlinear Problems 6.1 Introduction 6.2 Proofs 6.3 The Eigenvalue Problem

141 141 145 152

7

Systems with Hamiltonian Potentials 7.1 A Linking Theorem 7.2 Hamiltonian Elliptic Systems

159 159 167

8

Linking and Elliptic Systems 8.1 An Infinite-Dimensional Linking Theorem 8.2 Elliptic Systems

179 179 187

9

Sign-Changing Solutions 9.1 Linking and Sign-Changing Solutions 9.2 Free Jumping Nonlinearities

195 195 202

10 Cohomology Groups 10.1 The Kryszewski-Szulkin Theory 10.2 Morse Inequalities 10.3 The Shifting Theorem 10.4 Critical Groups of Local Linking 10.5 Computations of Cohomology Groups 10.6 Hamiltonian Systems 10.7 Asymptotically Linear Beam Equations

215 215 227 231 242 244 253 269

Bibliography

287

Index

317

Preface Since t h e birth of t h e Calculus of Variations, it has been realized t h a t when they apply, variational methods can obtain better results t h a n most other methods. Moreover, they apply in a very large number of situations. It was realized many years ago t h a t t h e solutions of a great number of problems are in effect critical points of functionals. In this volume we present some of the latest research in the area of critical point theory. Many new results have been recently obtained by researchers using this approach, and in most cases comparable results have not been obtained by other methods. We describe these methods and present t h e newest applications. In a typical application, one first establishes t h a t the solution of a given problem is a critical point of a functional G{u) on an appropriate space, i.e., a "point" in t h e space where G\u) = 0. Finding t h e points where t h e derivatives vanish is t a n t a m o u n t to solving the problem. T h e main difficulty is finding candidates. In this connection, one can use "geometrical" considerations. But geometrical considerations do not involve derivatives, and usually the best they can produce are Palais-Smale sequences, i.e., sequences of t h e form G{uk) -^ a,

G\uk)

-^ 0.

T h e existence of such a sequence is not enough to produce a critical point. It is possible t h a t such a sequence is converging to infinity. However, if one can show t h a t the sequence has a convergent subsequence, then one indeed obtains a critical point. A functional t h a t has the property t h a t every Palais-Smale sequence for it produces a critical point is said to satisfy the Palais-Smale condition. W h a t is one to do if t h e corresponding functional does not satisfy the Palais-Smale condition? In the present volume, one of the purposes is to consider just this situation. T h e trick here is to find bounded Palais-Smale sequences directly from t h e linking geometry. In most cases such sequences

X

PREFACE

produce critical points. One might think that such methods are severely restricted. However, the number of such methods and applications found here should convince anyone otherwise. It is surprising that so much has been accomplished under this handicap resulting in new variational methods. Another purpose of this book is a description of the so-called topological method. We present a new Morse theory which satisfactorily fits strongly indefinite functionals. We include such topics as extrema, even functionals, weak and double linking, sign-changing solutions, Morse inequalities, and cohomology groups. The applications we describe include Hamiltonian systems, Schrodinger equations and systems, jumping nonlinear it ies, elliptic equations and systems, superlinear problems and beam equations. The book is organized as follows. In Chapter 1, we provide some prerequisites for this monograph. We collect some knowledge of degree theory, Sobolev space and so forth. Basically, these theories are essentially known and readily available in many books. Welltrained readers may skip this chapter. In Chapter 2, we present some theorems concerning functionals which are bounded below on Banach spaces or Finsler manifolds. Chapter 3 is devote to critical point theory on even functionals. Some variants of the fountain theorem will be established without (PS) type assumptions. Applications to Schrodinger equations and Dirichlet boundary value problems will be given. We will show readers how to get infinitely many solutions. In Chapter 4, we establish a weak infinite-dimensional linking theorem. It not only unifies the classical results but also gives us more information. This abstract theory works perfectly for some PDEs and ODEs with pure continuous spectrum. Therefore, applications will be considered mainly on homoclinic type solutions of asymptotically linear Hamiltonian systems and Schrodinger equations, superlinear Schrodinger equations with zero as a point of the spectrum or with critical Sobolev exponents. In particular, Schrodinger systems depending on time will be discussed. Chapter 5 concerns twin critical points resulting from double linking. Roughly speaking, if A links 5 , does B link A? Can they yield two different critical points without the (PS) type compactness conditions? We will give positive answers. Applications on eigenvalue problems and Dirichlet elliptic equations with jumping nonlinearities will be studied.

PREFACE

xi

In Chapter 6, we solve elliptic semilinear boundary value problems in which the nonlinear terms are quite weak super-linear. That is, the nonlinearities need not satisfy a superquadracity condition of the Ambrosetti-Rabinowitz type. Because of this, we are able to include more equations than hitherto permitted. Some new tricks will be seen. In Chapter 7, we assume that A links B. Let Bi and B2 be two linear bounded invertible operators. We describe the situation in which the values of the functional H are separated by BiB and B2A. Then BiB and B2A become much more complicated. We prove the existence of a critical point of H without assuming (PS) type conditions. This theory is applied to some special elliptic systems. In Chapter 8 we prove an infinite-dimensional linking theorem and apply it to elliptic systems with gradient type potentials. In Chapter 9, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A linking type theorem is established with the location of the critical point in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities. Under stronger conditions, we show that the existence of sign-changing critical points can be independent of the Fucik spectrum which usually is indispensable for such cases. In Chapter 10, a more advanced Morse theory will be introduced. We first present the W. Kryszewski-A. Szulkin infinite-dimensional cohomology theory and a new Morse theory associated with it. Then we develop some methods of computing the cohomology critical groups precisely. Applications to Hamiltonian systems and beam equations will be considered. The present monograph is based on results obtained by ourselves or through direct cooperation with other mathematicians such as S. Li, A. Szulkin, Z. Q. Wang and M. Willem. It is not intended to be complete. The materials covered in this book are presented at a level suitable for advanced graduates and Ph. D. students following the development of new results, or anyone who wishes to seek an introduction to critical point theory and the study of differential equations by variational and topological methods. The chapters are designed to be as self-contained as possible.

xii

PREFACE

Both Zou and Schechter thank the University of Cahfornia at Irvine for providing a favorable environment during the period 2001-2004 in which the first version of this book was written. Both authors wish to thank the NSF, NSFC (No. 10571096 & No. 10001019) and SRF-ROCS-SEM for supporting much of the work that led to this book.

Wenming Zou Tsinghua University, Beijing Martin Schechter University of California at Irvine

Chapter 1

Preliminaries In this chapter, we present some classical results on nonlinear functional analysis and partial differential equation. Some of them are well known and we shall omit their proofs. For others, although their proofs may be found in many existing books, we make no apology for repeating them.

1.1

Partition of Unity in Metric Spaces

Assume {E,d) is a metric space with a distance function (i(-, •). Let A C E and let 11 be a family of open subsets of ^ . If each point of A belongs to at least one member of 11, then 11 is called an open covering of A. Definition 1.1. Assume that H is an open covering of a subset A of E, then n is called locally finite if for any u ^ A, there is an open neighborhood U such that u e U and that U intersects only finitely many elements of li. The following result is due to A. H. Stone [347]. Proposition 1.2. Any metric space {E,d) is paracompact in the sense that every open covering li of E has an open, locally finite refinement Q, i.e., 6 is a locally finite covering of E and for any Vi of Q, we can find a Ui of li such that Vi C Ui. Proposition 1.3. Assume that {E, d) is a metric space with an open covering n . Then li admits a locally finite partition of unity {\i}i^j subordinate to it satisfying: (1) Xi : E ^ [0,1] is Lipschitz continuous; (2) {Vi}i^j is a locally finite covering of E, where Vi = {u ^ E : \i{u) 0}, J is the index set;

^

2

CHAPTER 1. (3) for each Vi, there is a Ui eli

PRELIMINARIES

such that Vi C Ui]

(4) E i e J ^ i W = l ' V u e ^ . Proof. Since {E, d) is a metric space with an open covering 11, by Proposition 1.2, there is an open, locally finite refinement B, i.e., B is locally finite and for any Vi of B, we can find a [/^ of 11 such that Vi ^Ui. We define pi{u) = d{u,E\Vi),

i G J.

Then pi is locally Lipschitz. Let

Then {Xi}i^j is what we want. This proves the theorem.

1.2

D

Sobolev Spaces

Let O be an open subset of R ^ , AT G N. Define 1/^(0) := {i^ : O ^ R is Lebesgue measurable, ||I^||LP(Q) < oo}, where \\U\\LP{Q)

= y

\u\Pdxj

,

l 0 or m = oo). Let C"^(0) be the set of functions in C"^(0) all of whose derivatives of order < m have continuous extension to fl. Definition 1.5. Fix p G [1, +oo] and A: G N U {0}. The Sobolev space

consists of all u : Q ^ H which have a^^- weak partial derivatives D^u for each multiindex a with |Q^| < A: and D^u G 1/^(0). If p = 2, we usually write H^{n) = iy^'2(0), Note that H^{n) which agree a.e.

= L'^{n).

A: = 0 , 1 , 2 , . . . .

We henceforth identify functions in ly^'^(O)

Definition 1.6. If u e VF^'^(O), we define its norm to be (

/

\ i/p vjv

Definition 1.7. We denote

as the closure ofC^{Q) in VF^'^(O) with respect to its norm defined in Definition 1.6. It is customary to write

and denote by II~^{Q) the dual space to

IIQ{Q).

CHAPTER 1.

PRELIMINARIES

The following results can be found in L. C. Evans [147]. P r o p o s i t i o n 1.8. For each A: = 1, 2 , . . . and I < p < +oo, the Sobolev space f VF^'^(O), II • ||vi/fc'P(Q)) ^-5 0. Banach space and so is H^{Q),HQ{Q)

WQ'^{Q).

In particular,

are Hilhert spaces.

cY. Definition 1.9. Let (X, || • ||x) CL^id (F, || • \\Y) he two Banach spaces, X We say that X is continuously imbedded in Y (denoted by X ^^ Y) if the identity id : X ^ Y is a linear bounded operator, that is, there is a constant C > 0 such that \\U\\Y < C||i^||x for all u e X. In this case, constant C > 0 is called the embedding constant. If moreover, each bounded sequence in X is precompact inY, we say the embedding is compact, written X ^^^^ Y. Definition 1.10. A function u : Q C R ^ ^ H is Holder continuous with exponent 7 > 0 i/ 1(7) .-

\u(x) — u(y)\ ^ < 00. \x-y\-r

sup

^ ^

Definition 1.11. The Holder space C'^''^(f2) consists of all functions u G C''(f2) for which the norm \\u\\cK.m:=

Y,

P"«llc(n)

\a\ 0,m > 0 such that \g{x,t)\ < (i|t|^i/^S \h{x,t)\ < m|t|^2/^^

10

CHAPTER 1.

PRELIMINARIES

Define Biu = g{x,u),

u G I/^'(0);

B2U = h{x,u),

ueLP^{n).

Then by Lemma 1.20, Bi is a bounded and continuous mapping from L^^^Q) to L^^ (^), ^ = 1, 2. It is readily seen that B := Bi-\-B2 is a bounded continuous mapping from H to S. D The fohowing theorem and its idea of proof are enough for us to see that the functionals encountered in this book are of C^. Theorem 1.22. Assume a > 0,p > 0. Let f{x,t) on ft xH satisfying

(1.10)

be a Caratheodory function

1/(^,01 < «IC + ^l^r. V(x,t) G O X R,

where a, 6 > 0 and ft is either bounded or unbounded. Define a functional I{u) := / F{x,u)dx,

where F{x,u) = /

JQ

f{x,s)ds.

JO

Assume {E, \\ • ||) is a Sobolev Banach space such that E ^^ L^^^iQ) E ^ L^+i(0). Then I G C^{E,Ii) and I'{u)h = I f{x,u)hdx,

and

\/h G E.

JQ

Moreover, if E ^^^^ L^^'^^E ^^^^ L^^^, then I' : E ^ E' is compact. Proof. Since E ^^ L^^^(Q) and E ^^ 1/^+^(0), we may find a constant Co > 0 such that (1.11)

||u;|| 0, there exists a finite-dimensional subspace E^ of E and a bounded continuous mapping I^ : M ^ E^ such that s u p ||/(l^) -In{u)\\ uEM

< S.

1.4. TOPOLOGICAL DEGREES

15

Let ^ be a Banach space, and let U C E he di bounded open subset. Let I : U :^ E he completely continuous and f = id — I. If p e E\f{dU), then by Theorem L32, there exists a finite-dimensional subspace E^ of E and a bounded continuous mapping 1^:1)^ E^ such that sup||/(^)-/^H|| 0 depending on r and i^o such that \\V{wi) -V{w2)\\

< p\\wi -W2\\,

y wi,W2 e

B{uo,r).

Let A:=

sup

||y||.

B(uo,r)

Then A < +00. Choose £ > 0 such that sp < l^sA < r. Consider the Banach space E := C([0,£],F) := {u : [0,^] ^ F is a continuous function} with the norm ||i^||^ := max^^[o,£] ||'?^(OII for each u e E. Let D := {u e E : \\u — uo\\^ < r}. Define a mapping F : E ^ E hj Fu := 1^0 + / V{u{s))ds, Jo

u e E.

For any u,w ^ D we have \\Fu-uoh
0}; (2) ip{B n dn) C dK^ := {x = (xi, X2,..., XN) eK^ :XN = 0};

The following proposition is due to M. D. Gilbarg-N. S. Trudinger [174, Theorems 6.6] (see also M. Struwe [352]). Proposition 1.43. Suppose that u G Hi^^{ft) such that —Au = f in ft with f G I/^(^), 1 < p < 00. Then for any O^ CC ft, we have \\U\\H^^P{Q^) < C{\\U\\LP^Q^ + | | / | | L P ( Q ) ) ,

where C depends on ft,ft\N,p. Assume in addition that ft is a C^'^ domain and that there exists a function uo G H'^'^{ft) such that u — uo ^ HQ'-^{ft). Then \\U\\H^^P{Q)

< C{\\U\\LP(Q)

+ ||/||LP(Q) +

||^0||if2,p(Q)),

where C depends on ft^N^p. The following proposition is found in M. D. Gilbarg-N.S. Trudinger [174, Theorems 6.14 and 6.19]. Proposition 1.44. Assume that ft is a Qk+2,a domain, f G C^'^(O). Then the Dirichlet problem —Au = f

in ft,

u=0

in dft

has a unique classical solution u G C^+^'^(0). The following proposition is also in M. D. Gilbarg-N.S. Trudinger [174, Theorem 9.15]. Proposition 1.45. Assume that ft is a C^'^ domain, f G L^{ft),p > 1. Then the Dirichlet problem —Au = f

in ft,

u=0

in dft

has a unique classical solution u G I^o^'^l^) H W'^^P{n). The following result is due to H. Brezis-T. Kato [66] (see also M. Struwe [352]).

22

CHAPTER 1.

PRELIMINARIES

L e m m a 1.46. Assume that ft is a domain of R ^ (N > 3) and that f : ft xH ^H is a Caratheodory function such that \f{x,t)\ where a G LfJ^\n).

< a{x){l + |i^|),

If u e HHifl)

a.e. x e Q,

is a weak solution of

—Au = f{x^u)

in O,

then u G L^^^(Q) for any q < oo. If u ^ HQ' (ft) and a G 1/^/^(0), then u G L^{Q) for any q < oo. Next, we give an example to illustrate when a weak solution becomes a classical solution. As we mentioned, it is not a matter of course. Theorem 1.47. Assume that Q is a bounded domain ofH^ {N > 2), Q is is a Caratheodory function such that (1) there exists a r G (0,1] such that f{x,t)

for any

GC^''^(OX [ - M , M ] , R ) ,

M

>0;

(2) there are C > 0 and 2 < p < 2* such that \f{x,t)\

< C ( 1 + |^|^-^),

(3) there exists a function fo{x) G L^{ft) lim

^— = fo{x)

a.e.xeQ;

such that uniformly for x G O.

Assume u G H^' (Q) is a weak solution of (1.23)

-/\u

= f{x,u)

inn,

u=0

on dn.

Then u must he a classical solution of (1.23). In particular, u G C^+^'^(0), where (3 = ar^~^^. Proof. Let

f{x,t) g{x,t)

t /o(x),

'

if t ^ 0, ift = 0.

1.7. WEAK SOLUTIONS

23

Then there are constants a > 0, 6 > 0 such that (1.24)

\g{x,t)\ < fo{x) + a + b\tf-^ < /o(x) + a + b\tf-^

for ah X G 0 , t G R. By assumption, u G HQ' ( O ) is a weak solution of (1.25)

—Au = g{x,u{x))u

in O,

i^ = 0

on dfl.

If N > 3, by Proposition 1.12, u G L'^*{n). By Lemma 1.20 and (1.24), g{x,u{x)) G 1/^/^(0). Then, Lemma 1.46 implies that u G 1/^(0) for ah 5 > 2. This is naturally true if A^ < 2. Noting the conditions (2)-(3) and using Lemma 1.20 again, we see that f{x,u{x))

GL^(O),

V5>2.

Choose 5 > 2 , 5 > p — 1 . By Proposition 1.45, the problem (1.26)

—Aw = f{x,u{x))

in O,

w=0

on dft

has a unique solution

w G w^'^n) n iy2'^(o), v^ =

^ > i, 5 > 2.

Since u is a weak solution of (1.23), we see from (1.26), that u = w.Uwe choose q = j ^ , then q > 2/{p - 1) if N > 2. By Proposition 1.12, u G I^o'^(^) implies that u G C^'^(O); here 1 — N/q = o^. Then we may find a M > 0 such that |^(x)|<M,

\u{x)-u{y)\

VXGO,

<M|x-7/|^,

x.yen.

Note that O is of class C^+^.Q^ with a G (0,1). Thus / satisfies condition (1) with k replaced by 0,1, 2 , . . . , A: - 1 (see M. D. Gilbarg-N. S. Trudinger [174, Lemma 6.35] and W. Lu [245, Theorem 7.5.4]). Hence, there exists a C > 0 such that + \u-vn \f(x,u)-fiy,v)\{u)),ue

E.

The continuity of P{u) implies that there exists an open neighborhood U{u) of u such that a\\P{w)\\ < {P{w),(l){u)), w e U{u). Then we get an open covering {U{u)} oi E. By Proposition 1.3, there is a locally finite refinement {Vi\i^j and a locally Lipschitz continuous partition

26

CHAPTER 2. FUNCTIONALS

BOUNDED

BELOW

of unity {Xi}iej subordinate to this refinement. For each i e J, Vi C U{ui) for some ui. Define

Then V : E ^ E is locahy Lipschitz continuous. By Proposition 1.3,

\\V{w)\\aJ2Mw)\\Piw)\\=a\\Piw)\\. ieJ

D Notes and Comments. Many books and papers have addressed the existence of the pseudo-gradient vector field which was apphed directly to prove miscellaneous deformation theorems. For examples, see P. Bartolo-V. BenciD. Fortunato [28], V. Benci-P. H. Rabinowitz [55], K. C. Chang [95, 96] (on a Finsler manifold), Y. Du [143], M. R. Grossinho-S. A. Tersian [176], J. Mawhin-M. Willem [252], L. Nirenberg [265], P. Rabinowitz [293], M. RamosC. Rebelo [298], M. Schechter [310], M. Struwe [352] and M. Willem [376, 377].

2.2

Bounded Minimizing Sequences

Let {E, II • II) be a Banach space and / G C^(^,R). Theorem 2.2. Assume that there exist R > r > 0 such that m := inf / = inf / > —oo, BR

B,

where BR := {U ^ E : \\U\\ < R}. Then there exists {un} C BR such that I{un) -^ m,

I'{un) ^ 0

as n ^ oo.

Proof. Let D{R,e) = {u e BR : I(u) < m + e}. Then

inf

||/'(^)|| = 0

for all £ > 0. Otherwise, there would exist an SQ > 0 such that ||/^(i^)|| > £o/{R-r) wheni^ G D{R, SQ). Let u G D(r,£o/2)(^ 0). By Lemma 2.1, there is a V{u) : E :={ue E : r{u) ^ 0} ^ ^ such that 11^(^)11 < 1 ,

{V{u)J'{u))>\\\I'{u)l

yueE.

2.2. BOUNDED MINIMIZING SEQUENCES

27

Moreover, y is a locally Lipschitz continuous map. Let a{t,u) be the solution of the Cauchy initial value problem

a{0,u) =ue

D{r, So/2).

Then ||cr(t,i^)-1^11 < / \\a\s,u)\\ds 0 for all u e E. Assume that one of the following conditions holds. (A) For any /3 > 0,

sup {H'ull :ue

E with H{u) < (3} < +oo.

(B) For any /3 > 0,

sup {H'ull :ue

E with J{u) < (3} < +oo.

Theorem 2.3. Assume that either (A) or (B) holds and that Ix is bounded below for each A G A. Then for each X e A, there exists a sequence {un} such that s u p ||l^n|| < oo,

Ix{Un)

^

Mx

'•= i n f / A ,

Ixi^n)

^ 0 ,

aS Tl ^

OO.

28

CHAPTER 2. FUNCTIONALS

BOUNDED

BELOW

Proof. We only prove the first case. Note that the mapping X ^ Mx is concave with respect to A G A. Therefore, it is Lipschitz continuous on each closed subinterval of A. For A G A, we choose a closed subinterval AA C A containing A as an interior point. Then, there exists a constant Ai^ > 0 depending on A such that

\Mx-My\<M'^\X-X'\, Choose A„ e (A, 2A) n

AA,A„

VA'eA^.

-^ A as n ^

Then l - ^ - - - ^ - '
1\\I',{U)\\ > ^£0.

dfi (t u) Now we consider t h e initial value problem — - ^ — = —Vx{r]) with ?^(0, u) = u for each u ^ E (note t h a t ^ vanishes on an open set containing t h e points where / ^ = 0)- It is well known t h a t there exists a unique solution r]{t,u) for t >0. Moreover, Mx

< Ix{v{t.un))

< lx{r]{0,un))

< IxA^n)

< Mx^ + (A - A,) < A^A + ^

for n large enough. Consequently, = \\ [ dr]{s,Un)\\< Jo

\\r]{t,un)-uj

[ Jo

\\V^{r]{s,Unmds \\v - w\\h{\\v - w\\)

for all u G Ei^v^w G ^2, then we have the following results. (1) There exists a continuous function (j) : Ei ^ E2 such that I{u -\- (j){u)) = min I{u + v). veE2

Moreover, (j){u) is the unique member of E2 such that {I'{u^(j){u)),v)

=0,

\JveE2.

(2) The functional J : Ei ^ H defined by J{u) = I{u-\- (j){u)) is of class C^ and {J\u),v) = {I\u^d^{u)),v), yu.veEi. (3) An element u e Ei is a critical point of J if and only if u -\- (j){u) is a critical point of L Proof. (1) For each u G ^ 1 , define H^ : E2 ^ K by Hu{v) = I{u + v). By the assumption (2.1), Hu is of C^ and has at most one critical point. We claim that Hu is coercive. Note that Hu{v) = Hu{0)^

[ Jo

{Hu{sv),v)ds

>H^{0)-\\H:,m\\v\\+

f\\\v\M\sv\\)ds. Jo

By the hypotheses on /i, we may choose R large enough such that h{\\sv\\) > 8||i:f;(0)|| uniformly for \\v\\ >R,se

[1/2,1].

2.3. AN APPLICATION

33

Hence, Hu{v)>Hu{0)^\\vl which imphes that Hu{v) ^ oo as H^;!! ^ oo. Next, we show that Hu is convex. For given v^w G ^ 2 , define C{s) =

Hu{v^s{w-v)).

For 0 < a < f3 < 1, by (2.1), it is easy to show that

r(/3)-e'(«)>0. This means that ^ is convex in 5, and consequently, H^ is convex in v. Combining the above arguments, we see that Hu has a unique minimizer (l){u) G E2 with Hu{(j){u)) = min{/(i^ -^ v) : v ^ E2]. Therefore, we have (2.2)

{I'{u^(l){u)),w)={),

\IweE2.

To show that (j){u) is continuous in u^ we assume on the contrary, that there are £0 > 0 and Uk ^ u SiS k ^ 00 such that \\(l){uk) -(/>(^)|| >£o. Let P be the projection from E to ^2- By (2.2), we see that \\Pr{uk (/>('^))|| ^ h{£o/2) if A: large enough. Therefore, h{eo)muk)

- c^{u)\\

< {I\uk + (/)(^fe)) - I\uk + (/)(^)), (/)(^fe) - 0(^))

< (-r(^fe + H^)), (i){uk) - (i){u)) X\\v — w\\'^ — / a{v — w)'^dx JQ

>{X-T^)\\v-Wf. By Lemma 2.7, there exists a mapping 6x : E^ ^ E- ^ E+ such that Ix{u^ ^u-^(t)x{u^ Moreover, (t)\{u^ ^u~)

^u-))=

min /A(^^ + ^ " + ^ ^ ) .

is the unique member of E^ such that

for ah V e E^. Define a functional JA : ^^ © ^ ~ ^ R by Jxiu"" + U-) = Ixiu"" + ^ - + (/)A(^^ + u-)).

2.3. AN APPLICATION

35

Then J is of class C^ and

for all u^ -\-u~,z e E^ ^E~. Moreover, u^ -^u~ is a critical point of J if and only if u^ ^ u~ ^ (l)\{u^ -^ u~) is a critical point of I\. Next, we claim that —I\ is bounded below on E~ ® E^. In fact, by condition (C) we see that Afe^^ < t{g{x,t) - ^ ( x , 0 ) ) < at^. Then G{x,t)=

/ 5^(x,5)—>-Afet2 + t^(x,0). Jo

5

^

Therefore, Ix{u) < \{\ - l)\\uf 2

- I ug{x,0)dx ^ - o o Jn

as 111^11 -^ oo. Hence, —Ix and —Jx are bounded below. Evidently, —Jx satisfies the other assumptions of Theorem 2.3. Therefore, for all A G A, there exists a ux such that —J^^{ux) = 0. This completes the proof of the theorem. D Notes and Comments. Lemma 2.7 was established in A. Castro [81]. Some applications of it can be found in A. Castro-J. Cossio [82] and M. Schechter [315]. Theorem 2.6 was given in M. Schechter-W. Zou [326]. Possibly, it can be proved by other methods such as the degree theory or the contraction mapping principle. We believe that Theorem 2.4 has far more extended applications. We would like to leave them to the readers.

Chapter 3

Even Functionals In this chapter we present some abstract theorems which concern the existence of infinitely many critical points for even functionals. The Palais-Smale type compactness condition is not necessary for the new results. By taking advantage of the abstract theorems, we study the existence of infinitely many large energy solutions for nonlinear Schrodinger equations and of infinitely many small energy solutions for semilinear elliptic equations with concave and convex nonlinear it ies.

3.1

Abstract Theorems

Let ^ be a Banach space with the norm || • || and let {Xj} be a sequence of subspaces of E with dimX^ < oo for each j G N. Further, E = ^j^-^Xj, the closure of the direct sum of all Xj. Set

and Bk := {ueWk:

\\u\\ < pk},

Sk := {ue Zk'. \\u\\ = r/e}, for Pk > Tk > 0. Consider a family of C^-functionals ^\ : E ^Yi $A(W) := J(w) - AJ(w),

oi the form:

Ae[l,2].

We make the following assumptions. (Ai) ^\ maps bounded sets into bounded sets uniformly for A G [1,2]. Moreover, ^ A ( - ' ^ ) = ^A('^) for all (A,i^) G [1,2] x E.

38

CHAPTERS.

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FUNCTIONALS

(A2) J{u) > 0 for all u e E; I{u) ^ 00 or J{u) ^ 00 as ||i^|| -^ 00, or (A3) J{u) < 0 for all u e E; J{u) -^ —00 as ||i^|| -^ 00. Let afe(A) :=

max

^x(u),

bk{X) :=

ueWk,\\u\\=Pk

inf

^x{u).

ueZk,\\u\\=rk

Define Cfc(A) := inf max ^x{j{u)),

where

jeTkueBk

Tk := {7 e C{Bk,E) : 7 is odd,7|a5, = id],

k>2.

Theorem 3.1. Assume that (Ai) and either (A2) or (As) hold. If bk{X) > a/c(A) for all A G [1,2], then Ck{X) > bk{X) for all X G [1,2]. Moreover, for almost every X G [1,2], there exists a sequence {u^{X)}'^^i such that sup ||i^^(A)|| < 00, ^xi^^W)

-^ 0 and ^xi^nW)

^ Cfc(A)

as n ^ 00.

n

Proof. We divide the proof into two cases. Case 1: Assume that conditions (Ai) and (A2) hold. We show that Ck{X) > bk{X) first. For each 7 G r^, let U^ := {u e Bk : ||7('^)|| < 7fe}- Then U^ is an open bounded symmetric neighborhood of 0 in W^. Let P^ : E ^ W^-i be the projection onto Wk-i. Then Pkj : dU^ -^ Wk-i is a continuous odd map. By the Borsuk-Ulam Theorem (cf. Theorem 1.30), there exists a i^ G dU^ such that Pkj{u) = 0, that is, 7(1^) G Z^, ||7(i^)|| = r^. Therefore, j{Bk) nSk j^9. This implies Ck{X) > bk{X). Furthermore, Cfc(A) < max ^A('^) ^ i^ax ^i(i^) := m/c, uEBk

uEBk

where ruk is a constant independent of A. By (A2), Ck{X) is nonincreasing with respect to A. Therefore, c^(A) := ——— exists for a.e. A G [1,2]. dX From now on, we consider those A where the derivative c^(A) exists. Let A^ G [1, 2], A^ < A, A^ -^ A, then there exists an n(A) such that (3.1)

- 4 ( A ) - 1 < " ' - ( y _ - ^ ^ ( ^ ) < -c-(A) + 1

forn>n(A).

Step 1. We show that there exists a sequence 7^ G Tk^m := m(c^(A)) > 0 such that ||7n('^)|| < ^^T. if ^A(7n('^)) ^ ^^(A) — (A — A^) for some u G B^-

3.1. ABSTRACT

THEOREMS

39

Indeed, let 7^ G Tk be such that sup^^^^ ^x^hn{u)) < Ck{Xn) + (A - A^). If ^x{ln{u)) > Ck{X) — (A — An) for some u e Bk, then

.. .NX -^(TnN) =
1 for 'U G t/t^ H J^{X,k). Therefore, we get an open covering {Uu}ueJ^{x,k) of ^(A, A:). Choose an open set UQ := ^ ^ ^ ( - 0 0 , Cfc(A) - (A - A^)). Then {Uu}ueTix,k) U t/o is an open covering of ^(A, k):={ueE:

\\u\\ < m(4(A)) + 4, ^ A M < Cfe(A) + £0}.

40

CHAPTERS.

EVEN

FUNCTIONALS

Hence, there exists a refinement {Nj}j^j such that Nj C Uu or Sj C UQ and a locahy Lipschitz continuous partition of unity {Pj}jeJi where J is the index set. Define UJJ{U) = uj{u) for Nj C Uu] ^j{u) = 0 for Nj C t/o, and set J\f := U^GJ^j- Then ^(A, k) C A/". Let ^*(A, k):={ueE:

^x{u) < Ck{\) - 2(A - A,)}

and . , ^ dist(^,>F*(A,A:)) "^^^^ • " dist(^,J^*(A,A:)) + dist(^,J^(A,A:))' ^^ Define a vector field l^(i^) := '4^{u)^-^j PJ{U)(JOJ{U) : J\f ^ E and consider djTi

the following Cauchy problem — = —V{r]) and r]{0,u) = u for u ^ E with ll^ll < m(4(A)), Cfe(A) - (A - A,) < ^x{u) < Ck{X) + soThen V is locally Lipschitz continuous and for any u as above, there exists a unique solution ?^(-, u). Noting that for any u G J^£Q{X, A:), we have that either 2 2 cjo = 0 or \\iOn\\ = ||c(;(i^)|| = ,, ^, , ,,, < —. Therefore, V is bounded and ^

^

II^AMII

^0

2 11^(^)11 ^ —• It follows that r](',u) exists as long as it does not approach the £o

boundary of M. Moreover, {^'x{u),V{u)) > 0 , for ^ G AT, {^xi^), V{u)) > 1, for u G J^(A, k). Evidently, ^^^(^(^^^)) < Q. For each u G ^ ^ ^ ( - o o , Cfe(A) - 2(A - A^)), we have V{u) = 0 and ^x{r]{s,u)) < ^x{r]{0,u)) = ^x{u)' It follows that r]{s,u) G ^^^(-oo,Cfc(A) - 2(A - A^)) and V{r]{s,u)) = 0. Therefore, (3.2)

v{s,u) =u

for ue ^ ^ ^ ( - o o , Cfe(A) - 2(A - A^)).

On the other hand, since ^A is even, then we may choose r]{s, u) to be odd in u. Consider ?^(2£o,7n('^))- Then we claim that ?^(2£o,7n('^)) ^ ^k- Iii fact, for u G dBk,jn{u) = u, then r]{2£o,jn{u)) = r]{2£o,u). Since a/c(A) < c/c(A) —2(A —An), we have ^A('^) < «fe(A) < c/c(A) —2(A —A^) for i^ G ^ 5 ^ . By (3.2) r]{2£o,u) = 1^. Hence, 7^(260,7n('^)) = '^- Noting that 7n('?^) and r]{2£o,u) are odd, we see that 7^(2^0,7n('^)) is odd. Consequently, 7^(260,7n('^)) G F^. If ^A(7n('^)) < c/c(A) — (A —A^) for some u ^ B^, then r]{s,jn{u)) is well defined and (3.3)

$A(r/(2eo,7n(w))) 0; J{u) ^

for A G [1,2].

00 as \\u\\ -^ 00 on any finite-dimensional

Further,

subspace

ofE; (B3) there exist pk > rk > 0 such that afe(A) := bk{X)

inf ^x{u) ueZk,\\u\\=pk

:=

max

> 0,

^A('^) < 0

ueWk,\\u\\=rk for all A G [1,2] and dfe(A):=

inf

$A(n)-0

ueZk,\\u\\ c > 0. Moreover, for every M > 0, meas{{x G R ^ : b{x) < M}) < oo, where meas denotes the Lebesgue measure in R ^ . (C2) / G C ( R ^ X R , R ) , | / ( X , ^ ) | < C ( 1 + 1^1^-1) for a.e. x G O and ah ^ G R,

where 2 < p < 2*, / ( x , u)u > 0 for all i^ > 0. fix UiU

(C3) liminf—-^. \u\^oo

\u\^

> c > 0 uniformly for x G R ^ , where /i > 2 is a

constant. (C4) lim li^O

^— = 0 uniformly for x G R ^ ; U

^— is a nondecreasing funcU

tion of u for every x G R ^ . (C5) / ( x , u) is odd in i^ for any u ^ E and x G R ^ . T h e o r e m 3.3. Assume that (Ci)-(C^) hold, then problem (S) has infinitely many solutions {uk} satisfying - I {\\j Uk\^ ^b{x)u\)dx 2 JRN

— I

JRN

F{x,Uk)dx ^ 00

as A: ^ 00,

where F(x, i^) = J^ / ( x , 5)(i5. Let E :={ue

iy^'^(R^, R) : /

(| V ^P + b{x)u^)dx < 00}.

Then ^ is a Hilbert space with the inner product (i^, v) =

{\/u ' \/v -\- b{x)uv)dx

and the norm ||i^|| = {u,uy^'^. Then obviously, E ^^ I/^(R^). By Propositions 1.13 and 1.16 (Gagliardo-Nirenberg Inequality), we see that ^-^L"(R^),

V5G [2,2*].

Without loss of generality, we assume A^ > 3.

3.2. HIGH ENERGY SOLUTIONS L e m m a 3.4. E ^ ^

45

L^(R^) for 2 < s < 2N/{N - 2).

Proof. Let {un} C ^ be a sequence of E such that Un ^ u weakly in E. Then {||i^^||} is bounded and u^ ^ u strongly in Lf^^(R^) for 2 < 5 < 2N/{N-2). We first show that u^ ^ u strongly in I/^(R^). It suffices to prove that ^n •= II'^nII2 -^ II'^112- Assume, up to a subsequence, that 5^ -^ 5. For any bounded domain O in R ^ ,

/ \Unfdx< [

\unfdx -^ 5'^

hence S > \\u\\2. Let A{R,M)

:= {x G n^\BR

: h{x) > M } ,

B{R,M)

:= {x G n^\BR

: 6(x) < M } .

Then

/

0, we have that u^dx = / II^\BR

JA{R,M)

u^dx -\- /

u^dx < s. JB{R,M)

Therefore, 11^112 = Il^lli2(5^) + ll^lli2(RA^\5^) > Jim^ Il^n||i2(5^) >6^

-s.

It means that 5 = ||i^||2- Finally, by Gagliardo-Nirenberg Inequality (see This completes Proposition LI6): i^^ ^ ^^ in L^(R^) for 2 < s < 2N/{N-2). the proof of the lemma. D Let ^(i^) = - | | i ^ f - /

F{x,u)dx,

ueE.

46

CHAPTERS.

EVEN

FUNCTIONALS

Then by Lemma 1.20, ^ G C^{E, R ) and J^ is compact, where we take J{u) = /

F{x^u)dx.

We shah prove Theorem 3.3 by finding t h e critical points of

^. We choose an orthonormal basis {cj} of E and define X^ : = R e ^ . Then Zk^Wk can be defined as in t h e previous section. Consider ^x : E ^ H defined by ^ A H := - l l ^ f - A

T h e n J{u)

> 0, /(i^) ^

/

F{x,u)dx:=I{u)-XJ{u),

A G [1,2].

oo as ||i^|| -^ oo, ^ A ( - ' ^ ) = ^ A ( ' ^ ) for ah A G [1, 2] and

1^ G ^ .

L e m m a 3 . 5 . Under the assumptions of Theorem 3.3, for each A: > 2, there exist A^ ^ 1 as n ^ oo, Ck >bk > 0 and { ^ n } ^ i C E such that

P r o o f . Evidently, by conditions (C2), (C3) and (C4), for any £ > 0, there exists a C^ such t h a t f{x^u)u > Cs\u\^ —£\u\'^ for any u. Therefore, it is easy to prove, for some Pk > ^ large enough, t h a t afe(A) : =

max ^A(^) < 0 ueWkMu\\=pk

uniformly for A G [1,2]. On t h e other hand, by (C4), for any £ > 0, there exists a Cg > 0 such t h a t \f{x,u)\ < £\U\-\-CS\U\P~'^ for any x G R ^ , i ^ G R . Let ak : =

sup \\u\\p, ueZk,\\u\\=i

then ak ^ 0 diS k ^ 00. Indeed, suppose t h a t this is not t h e case. Then there is an £0 and {uj} C E with Uj ± Wk--i, \\uj\\ = 1, \\uj\\p > SQ, where kj -^ 00 as j ^ 00. For any v e E, we may find a Wj G Wk^-i such t h a t Wj ^ i; as j ^ 00. Therefore, \{Uj,v)\ = \{Uj,Wj -V)\


l\\uf-^Ml-^\\u\\;

> \hf-c\\u\\i >

\hf-cal\\ur.

If we choose rk := (4cpQ^^)^/^^~^\ then for u e Zk with ||i^|| = r/c, we get that

^xiu)>iAcpalf/(^-''\\-^):=h. It follows that bkW '=

inf

>bk^oo

ueZk,\\u\\=rk

as A: ^ oo uniformly for A. Therefore, by Theorem 3.1, for a.e. A G [1,2], there exists a sequence {u^{X)}'^^i such that sup||«^(A)||

c;

F{x,Wn)dx

it follows that lim ^A^(^n^n) = oo. Obviously, t^ G (0,1). Hence, (^';,^(t^Z^),t^Z^) = 0 . Thus, /

{-f{x,tnZn)tnZn

- F{x,tnZn))dx

-^ OO.

^

JUN

By condition (C4), h{t) = -t^f{x^s)s

— F{x,ts)

is increasing in t G [0,1];

hence, -f{x,s)s — F{x,s) is increasing in 5 > 0. Combining these with the oddness of / and noting that ^n

/ JUN

{-f{x,Zn)Zn

- F{x,Zn))dx

= ^x^{Zn)

G [6fe,Cfe],

^

thus we see that

/ ^

\'^J\'^i^nZn)^nZn

-t^

yX^tfiZfi)juX

00.

This provides a contradiction.

D

Proof of Theorem 3.3. This is a straightforward consequence of Lemmas 3.5 and 3.6. D Notes and Comments. An important theory for getting high energy solutions is the well known symmetric mountain pass theorem (SMP, for short) based on the (PS)-condition and index theory, see e.g. A. Ambrosetti-P. Rabinowitz [19]). The readers may find some variants of it in M. Struwe [352]. There is an extensive literature concerning the existence of infinitely many

3.3. SMALL ENERGY SOLUTIONS

49

high energy solutions via SMP and the Fountain Theorem (cf. P. Rabinowitz [293], T. Bartsch [30], T. Bartsch-M. Wihem [48], M. Struwe [352], and also M. Willem [377], etc). In particular, in T. Bartsch-Z. Liu-T. Weth [38], S. Li-Z. Q. Wang [217] and W. Zou [391], sign-changing high energy solutions were obtained. We will address this topic later in this book. Several authors considered the existence of high energy solutions with a perturbation from symmetry. For instance, the special case —Ai^ = \u\^~'^u -\-p{x) in O with 1^ = 0 on dQ was first studied by A. Bahri-H. Berestycki [25] and M. Struwe [348] independently (see also A. Bahri [24] and A. Bahri-P. L. Lions [27]). In P. Rabinowitz [293, 296] and K. Tanaka [364, 365] (and also G. C. Dong-S. Li [139]), the authors considered a general case of perturbed elliptic equations. In [367], H. T. Tehrani considered the case of a sign-changing potential. By using the ideas of P. Bolle [59], C. Christine-N. Ghoussoub [101] also obtained some results on perturbed elliptic equations. Applications of the perturbation theory to Hamiltonian systems are given by A. BahriH. Berestycki [26], P. Rabinowitz [296] and Y. Long [239]. Basically, ah the papers mentioned above only concern the existence of the solutions. In M. Schechter-W. Zou [332], infinitely many high energy sign-changing solutions for perturbed elliptic equations with Hardy potentials were initially obtained. However, whether the symmetry can be cancelled completely is even today not adequately solved (see P. Rabinowitz [293, 296], M. Struwe [352] and M. Schechter-W. Zou [332]).

3.3

Small Energy Solutions

We consider the following elliptic equation with concave and convex nonlinearities: (D)

—Au = f{x,u)-\-g{x,u)

in O,

i^ = 0 on dft,

where O is a bounded smooth domain of R ^ , A^ > L (Di) f,g e C{Q X R, R) are odd in u. (D2) There exist cr,5 e (1, 2), ci > 0, C2 > 0, C3 > 0 such that ci|i^|^ < f{x,u)u

< C2\u\^ -\- csli^l

for a.e. x G O and i^ G R.

(D3) There exists p G [2,2*) such that |^(x, 1^)1 < c(l +|i^|^~^) for a.e. x e ft and 1^ G R. Moreover, lim g{x,u)/u = 0 uniformly for x G O. (D4) Suppose that one of the following conditions holds (1) \im.\u\^00 g{x,u)/u

= 0 uniformly for x G O;

50

CHAPTERS.

EVEN

FUNCTIONALS

(2) lim|^|^oo^(x,i^)/i^ = —oo uniformly for x e Q. Furthermore,

fix u) ^—

and

^— are decreasing in u for u large enough; u (3) \im.\u\^oo g{x,u)/u = oo uniformly for x G O; g{x,u)/u is increasing in u for u large enough. Moreover, there exists a > maxjcr, 5} such that _. . ^ g(x,u)u — 2G(x,u) ^ .r i r ^ limmi ^—• > c > 0 umiormly tor x G O.

We let F and G denote the primitive functions of / and g respectively. Consider the example / ( x , u) + g{x, u) = X\u\'^~'^u + fi\u\P~'u. which satisfies (Di)-(D4). For the case of 0 < A < < fi = 1,1 < q < 2 < p < 2*, this problem was solved by A. Ambrosetti-H. Brezis-G.Cerami [15]. They also raised an open problem about the existence of infinitely many solutions for all A > 0. This open problem was studied in T. Bartsch-M. Willem [48]. Theorem 3.7 is an improvement and generalization of the results in [48]. Another example is f{x,u)

=i^|i^|^~^ln(2 + |i^|),

a G (1,2);

g{x,u) =/ii^ln(l + |i^|).

Then (Di), (D2), (D3) and (D4)-(2) hold if /i < 0; (Di), (D2), (D3) and (D4)-(3) hold with o^ = 2 if /i > 0. If we choose g{x,u) = u'^ for |i^| < 1; g{x,u) = c|^|-i/2ln(l + 1^1) for |^| > 1, then (Di), (D2), (D3) and (D4)-(l) hold. Theorem 3.7. Assume that (DiJ-fD^) hold. Then equation (D) has infinitely many solutions {uk] satisfying ^{uk) '•=-\\uk\\^ — I F{x,Uk)dx — I G{x,Uk)dx ^ 0~ 2 JQ JQ

as k ^ 00,

where \\u\\ = (J^ | y i^p(ix)^/^.

We choose an orthonormal basis {cj} of E := HQ{Q). Set Xj = Re^, W^ ®j=i-^j5 ^n = ®^nXj. Consider a family of C^-functionals: ^^[u) := -\\u\\^ 2

/ G{x,u)dx - X / F{x,u)dx JQ

JQ

:= I{u) - XJ{u),

3.3. SMALL ENERGY SOLUTIONS

51

where A G [1,2]. Then J{u) > 0 and J{u) ^ oo as ||i^|| ^ oo on any finitedimensional subspace. Let n > k > 2. Lemma 3.8. There exist Xn -^ l,u{Xn) G Wn such that ^xJwA^iK))

= 0,

^A.(^(An)) ^ Ck

as n ^ oo, where Ck G [(i^(2), 6^(1)]. Proof. We win apply Theorem 3.2. By (D3), for any £ > 0, there exists a Cs such that \G{x,u)\ < £\u\'^ -\- CS\U\P. Therefore, for ||i^|| small enough, ^ A H > ihW^ - c||i^||^ - c||i^|||. Assume a < 5 smd let ak{cr) :=

sup

||^||^, ak{S) :=

ueZk,\\u\\=i

sup

H^H^.

ueZk,\\u\\=i

Then ak{(j) -^ 0, ak{S) ^ 0 as A: ^ 00. For ll^ll := Pfe := (8c 0 and i^ G R. This provides a contradiction. If (D4)-(3) holds, then we have that

oo < c / \u{Xn)\'^dx^

[-g{x,u{Xn))u{Xn)-G{x,u{Xn))]

dx,

3.3. SMALL ENERGY SOLUTIONS

53

which imphes that / i -g{x, u{Xn))u{Xn) — G{x, u{Xn)) ) dx ^ oo. However, by the property of i^(A^), we have that c / ( -g{x, u{Xn))u{Xn) - G{x, u{Xn)) ] dx - c Jn v2^ / -

2

\ 2^*^^' ^(^ri))u{Xn)

+ ic /

< An /

\u{XnTdx

f -f{x,

-]-C

u{Xn))u{Xn)

-g{x,u{Xn))u{Xn)

- G{X, u{Xn)) I

\u{Xn)Vdx

- F{x,

j dx - \ c

u{Xn))

f

\u{Xn)\'dx

j dx

- G{x,u{Xn)) ] dx

2 such that 0 < jG{x,u)

< g{x,u)u,yu

G R\{0} and a.e. x G O,

where g is the nonlinear term and G is the primitive function of g; ft C R is bounded or unbounded. The coercivity condition was also used by W. OmanaM.Willem in [268] for Hamiltonian systems and by D. G. Costa in [106] for elliptic systems. In [268], with the aid of the superquadraticity condition, infinitely many homoclinic solutions were obtained if the system is odd. In [106], the existence of one solution was studied. In Z. Q. Wang [375], the author considered the effect of concave nonlinearities for the solutions of nonlinear boundary value problems such as Dirichlet (and Neumann) boundary value problems of elliptic equations. Infinitely many small energy solutions were obtained by different methods. His theoretical tools are D. C. Clark's theory for functionals bounded below (cf. D. C. Clark [102] and also H. P.

54

CHAPTERS.

EVEN

FUNCTIONALS

Heinz [179]), the Fountain Theorem of T. Bartsch-M. Willem [48] and M. Willem [377], and the trick of modifying the nonlinear term. In [375], Hamiltonian systems and wave equations were studied also. In N. Hirano [184], the author got infinitely many small energy solutions for sublinear equations by using relative homotopy groups. Multiplicity results for some nonlinear elliptic equations can also be found in A. Ambrosetti-J. Azorero-I. Peral [11] and A. Ambrosetti-J. Garcia Azorero-I. P. Alonso [17]. In particular, in S. Li-Z. Q. Wang [217], sign-changing small energy solutions were obtained. Theorems 3.3-3.7 of the present chapter were obtained by W. Zou in [385].

Chapter 4

Linking and Homoclinic Type Solutions In this chapter, we first prove a weak finking tfieorem wfiicfi, to some extent, unifies the classical linking theorems. Moreover, it produces a bounded Palais-Smale sequence for a non-even functional. Applications will be given on the existence of homoclinic orbits for Hamiltonian systems and solutions to Schrodinger equations.

4.1

A Weak Linking Theorem

Let ^ be a Hilbert space with norm || • || and having an orthogonal decomposition E = N Q M, where A^ C ^ is a closed and separable subspace. Since N is separable, we can define a new norm \v\w satisfying \v\w < ||'^||, ^ v G N such that the topology induced by this norm is equivalent to the weak topology of N on bounded subsets of N. For u = v -\- w G E = N ® M with V e N,w e M.v^e define \u\l^ = \v\l^ + ||'"^|P, then \u\w < \\u\\, y u e E. In particular, if u^ = v^ -\- w^ is | • 1^^ - bounded and u^ -^ u, then v^ ^^ v weakly in N, Wn ^ w strongly in M, Un ^ v -\- w weakly in E. Let Q C N be a II • 11-bounded open convex subset, po ^ Q be a fixed point. Let F be a I • I ^-continuous map from E onto N satisfying (i) F\Q = id; F maps bounded sets to bounded sets; (ii) there exists a fixed finite-dimensional subspace EQ of E such that

F{u -v)-

{F{u) - F{v)) cEo,\/v,ue

E;

(iii) F maps finite-dimensional subspaces of E into finite-dimensional subspaces of E;

56

CHAPTER

4. LINKING

AND HOMOCLINIC

TYPE

SOLUTIONS

Set A := dQ, B := F~^(po), where dQ denotes the || • ||-boundary of Q. There are many examples. E x a m p l e 4 . 1 . Let N = E-,M = E+, then E = E' © ^ + and let Q := {u e E~ : ||i^|| < i?},po = 0 G Q. For any u = u~ Qu^ e E, define F : E ^ N by Fu := u~^ then A := dQ^B := F~^{po) = E^ satisfy the above conditions. E x a m p l e 4 . 2 . Let E = E' ® E+, z^ G E+ with \\ZQ\\ = 1 , ^ + = R Z Q © E^. For any u ^ E, we write u = u~ ® sz^ © w^ with u~ G E~,s G Il,w~^ G (E-^Kzo)^ =Ef. LetN := E'^HZQ. For R > 0, let Q := {u :=u-^szo : 5 G R + , i ^ ~ G ^ ~ , ||i^|| < i?},po = 50^0 ^ Q, So > 0. Let F : E ^ N be defined by Fu := i^~ + ||5Zo+t^^||^o, then F , (3,po satisfy the above conditions with B = F~^{soZo)

= {u : = szo^w^

: s> 0,w^

G E^, \\szo ^w^\\

= so}.

In fact, according to the definition, F\Q = id and F maps bounded sets into bounded sets. On the other hand, for any u,v G E, we write u = u~ -\szo -\- w~^,v = v~ -\- tzo -\- w^, then F{u) = u~ ^ \\szo + i(;+||zo, F{v) = Therefore, v~ + \\tzo -^w^\\zo^ F(u — v) = u~ —v~ ^ \\{s — t)zo -\-w~^ —W^WZQ. F{u-v)-{F{u)-F{v)) = (^\\{s-t)zo^w^ C Hzo := Eo

For ^ eC\E,Il),

(4.1) r :=

-wtW

- \\szo ^ w^W ^ \\tzo ^

(a one-dimensional

wt\\)zo

subspace).

define h : [0,1] X Q ^ E is \ ' |^-continuous. For any (50,1^0) ^ [0,1] x Q, there is a | • |^-neighborhood Ui^s^^^^^ such t h a t {u - h{t,u) : {t,u) G U^so.uo) n ([o/l] X Q)} ^ ^ / m . /i(0,i^) =u,^{h{s,u)) < ^{u),\Ju G Q\.

Then F ^ 0 since id ^T. We shall always use Efin to denote various finitedimensional subspaces of E whose exact dimensions are irrelevant and depend on (50,1^0)- A variant weak linking theorem is T h e o r e m 4 . 3 . Let the family

of C^-functionals

^x{u):=I{u)-XJ{u), Assume

the following

conditions

hold.

( ^ A ) have the VAe[l,2].

form

4.1.

A WEAK

LINKING

(a) J{u) > 0 , V ^ G ^ ; ^ i

THEOREM

57

:=^.

(h) I{u) -^ oo or J{u) -^ oo as \\u\\ -^ oo. (c) ^\ is\-\w -upper semicontinuous; ^ ^ is weakly sequentially on E. Moreover, ^\ maps hounded sets into hounded sets.

continuous

(d) s u p ^ A < i n f ^ A , V A G [1,2]. A

B

Then for almost all A G [1,2], there exists a sequence {un} such that sup 11^^II < oo,

^ A K ) -^ 0,

^ A K ) -^ Cx;

n

where C\ := inf sup ^ A ( ^ ( 1 , ' ^ ) ) ^ [inf ^ A , s u p ^ ] .

P r o o f . We shall prove the theorem step by step. Step 1. We show t h a t C\ G [inf ^ A , sup ^ ] . Evidently, by t h e definition of C\,

C\ < sup ^ A ( ' ^ ) < sup ^i(i^) = sup ^{u) uEQ uEQ uEQ


inf^ ^ A for all A G [1, 2], we have to prove t h a t /i(l, Q) H 5 ^ 0 for ah h eV. By hypothesis, t h e m a p F / i : [0,1] x Q ^ AT is | • |^continuous. Let K := [0,1] x Q. Then K is | • l^^-compact. In fact, since K is bounded with respect to both norms | • |^ and || • ||, for any (tn^Vn) G K , we may assume t h a t Vn -^ VQ weakly in E and t h a t tn ^ to e [0,1]. Then VQ e Q since Q is convex. Since on t h e bounded set Q C N, t h e | • |^-topology is equivalent to t h e weak topology, then Un -^ VQ. SO, K is I • l^-compact. By t h e definition of F, for any (SQ, i^o) ^ K, there is a | • |^-neighborhood Ui^s^^^^^ such t h a t {u-h{t,u) : {t,u) G U^so.uo)^^} ^ ^fin- Note t h a t , K C y^^s,u)^KU^s,u)' Since K is | • |^-compact, K C U]^^[/(5.^^.), (si^Ui) G K. Consequently, {u — h{t,u) : (t^u) G K} C Efin. Hence, by t h e basic assumptions (i)-(iii) on F , F{u-h(t,u) : (t,u) G K } C Efin dind {u-Fh{t,u) : (t,u) G K } C Efin. Then we can choose a finite-dimensional subspace Efi^ such t h a t po ^ ^ / m and t h a t Fh : [0,1] x (Q n Efin) -^ Efin- We claim t h a t Fh{t, u) ^ po for ah u G d{Q n Efin) = dQ n Efin, t ^ [O51]- To see this, assume t h a t there exist to G [0,1] and i^o ^ dQ HEfin such t h a t Fh{to, UQ) = po, i-e., h{to,uo) G 5 . It follows t h a t ^i(i^o) ^ ^i(/^(^o,'^o)) ^ inf^ ^ 1 > sup^g ^ 1 , which contradicts t h e assumption (d). Thus, our claim above is true. By the homotopy

58

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

invariance of the Brouwer degree, we get that deg(F/i(l,.),Qn^/,^,Po) = deg{Fh{0,'),QnEf,r^,Po) = deg{id,QnEfin,Po) = 1. Therefore, there exists UQ ^ Q r\ Efin such that Fh{l,uo)

= po-

Step 2. Similar to step 1 in the proof of Theorem 3.1, we may consider only those A G [1,2] where C^ exists and use the monotonicity method. Let \n G [1, 2] be a strictly increasing sequence of such points satisfying A^ -^ A. Then there exists n(A) large enough such that (4.2)

- C ; - 1 < ^ ^ - ~ ^ ^ < - C ; + 1 for

n>n(A).

A — An.

Step 3. There exists a sequence h^ ^V^k := k{\) > 0 such that ||/i^(l,i^)|| < k if ^x{hn{l,u)) > Cx — {X — An). This is an analogue of step 1 in the proof of Theorem 3.1. In fact, by the definition of CA^, let /i^ G F be such that (4.3)

sup ^xAhn{l,u)) ueQ

< Cx^ + (A - A^).

Therefore, if ^x{hn{^,u)) > CA — (A — A^) for some u ^ Q, then for n > n(A)(large enough), by (4.2) and (4.3), J{hn{l,u)) < - C ^ + 3, I{hn{l,u)) < Cx - A(:7^ + 3A. By assumption (b), \\hn{l,u)\\ < k := k{X). Step 4. By step 2 and (4.3), sup^x{hn{hu)) ueQ

< sup^xAhnihu)) ueQ

< CA + (2 - C'x){X - Xn).

Step 5. For £ > 0, define (4.4)

Te{X) :={ueE:\\u\\ 0 small enough, that (4.5)

i n f { | | $ ^ ( u ) | | : u e ^ e ( A ) } = 0.

Otherwise, there exists an SQ > 0 such that ||^^(i^)|| > SQ for all u G J^SQ{X). Let /in ^ F be as in Steps 3-4 and n be large enough such that A — A^ < ^o and (2 - C'^){X - A^) < ^o- Define (4.6)

J^:^{X) :={ueE:

\\u\\ < A: + 4, CA - (A - A^) < ^ A ( ^ ) < ^A + ^ o } .

4.1.

A WEAK

LINKING

Clearly, J^^^{X) C J^soW(4.7)

THEOREM

59

Consider

^*(A) :={ueE:

^x{u)

< CA - (A - A^)}

and J^*o(A) U j^*(A). Since 11^^(^)11 ^ ^o for u G J^^oW, there is a ^{u) G E with 11^(^)11 = 1 such t h a t {^'^{u),^{u)) _

p f ^

> -\\^'^{u)\\.

We let hx{u)

:=

o

for u e Tl^(A).

Then ($';,(«), /IA(U)) > 2 for U e ^ , ; ( A ) . Since ^'^

is weakly sequentially continuous, if {i^n} is || • ||-bounded and u^ -^ u, then Un ^ u in E. Hence {^^^{un),hx{u)) -^ {^^^{u),hx{u)) as n ^ oo. It follows t h a t (^^(•), hx{u)) is | • |^y-continuous on sets bounded in E. Therefore, there is an open | • |^y-neighborhood Afu of u such t h a t (^^('u), hx{u)) > 1 for 'u G Afu^u G ^ * Q ( A ) . On t h e other hand, since ^ A is | • |^-upper semi-continuous, j^*(A) is I • l^-open. Consequently, Afx := Wu - u G J^toW) U J^*(A) is an open cover of ^ * Q ( A ) U ^ * ( A ) . NOW we may find a | • | ^-locally finite and \-\w open refinement {Uj)j^j with a corresponding | • li^- Lipschitz continuous partition of unity {f3j)j^j. For each j , we can either find Uj G ^ * Q ( A ) such t h a t Uj C A/'ii^, or if such u does not exist, then Uj C JF*(A). In the first case we set Wj{u) = hx{uj); in t h e second case, we take Wj{u) = 0. Let [/* = Uj^jUj, then [/* is I • 1^ - open, and J^*Q(A) U J ^ * ( A ) C U*. Define

(4.8)

rA(^):=E/^^(^H(^)-

Then FA • ^ * ^ ^ is a vector field which has t h e following properties: (i) Yx is locally Lipschitz continuous in both || • || and | • 1^^ topology; (ii) (iii)

{^'^{u),Yx{u))>0,yueU*; ($^(u),FA(ti))>l,Vue^;„(A);

(iv) | F A ( U ) U < \\Yxiu)\\ < 2/eo for u e U* and all A e [1,2]. dfi (t u) Consider the following initial value problem — - ^ — = —Yx{r]) with ?^(0, u) = u for ah u G J^*(A) U J?='(A,£o), where J^*{X) is given by (4.7) and J^(A,£o) := {^ G ^ : ll^ll < A:, CA - (A - A,) < ^ A ( ^ ) < ^ A + ^0} (4.9)

C^;(A).

Then by Theorem L36, for each u as above, there exists a unique solution r]{t,u) as long as it does not approach the boundary of [/*. Furthermore, t -^ ^x{v{^^^)) is nonincreasing.

60

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

Step 6. We prove that r]{t,u) is | • |^y-continuous for t G [0, 2£o] and u G j^(A,£o) U j^*(A). For fixed to G [0,2£o] and i^o ^ ^(A,£o) U J^*(A), we see that (4.10)

r]{t,u)-r]{t,uo)

=u-uo^

/

[Yx{r]{s,uo))

-Yx{r]{s,u))jds.

Since the set A := ?^([0,2£o] x {'^o}) is compact and | • |^y-compact and Yx is I • l^y-locally | • l^^-Lipschitz, there exist ri > 0,r2 > 0 such that {u G E : infeGA \u-e\^ < r i } C t/* and \Yx{u)-Yx{v)\w < r2\u-v\w for any i^,v G A. Suppose that r]{s,u) G t/* for 0 < 5 < t. Then by (4.10), \r]{t,u) -r]{t,uo)\w 0, r4 > 0 such that A2 :={ue E : | i ^ - A i | ^ < rs} C t/*; \Yx{u)-Yx{v)\^ < r4\u-v\^

62

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

for all u^v e A2 and ^^(^2) C ^/m- Evidently, by the | • |^ continuity of and hn, there exists a | • |^-neighborhood [/? ^^ such that (4.17)

YX^T],

r]{t,hn{l,u))cA2

for t G [0,2£o] and i^ G ^^5^^^^)- For t G [1/2,1], note that hn{t,u) -r]{A£ot-

2£o,hn{l,u))

= hn{t, u) - K{1, u)^

i

Yx{r]{s, /i^(l, u)))ds,

from which we conclude by (4.17) that (4.18) {hn{t, u) - r]{4.sot - 2^0, /^n(l, u)) : (t, ^) G t/f^^^^^^ n ([1/2,1] x Q)} C Ef,nBy the definition of TJ* , 1^ — r]*{t^u) u-hn{t,u)^hn{t,u) u-hn{t,u)

-^hn{t,u)

-hn{2t,u),

t G [0,1/2],

-r]{4sot-2so,hn{l,u)),

t G [1/2,1].

Combining (4.11), (4.12) and (4.18), we get that {u - r]%t,u) : (t^u) G %^^^^^ n ([0,1] x Q)} C ^ / . . , which implies that 7?* G T, where %o,uo) = ^lso,uo) ^ ^(so,uo) ^^ %o,uo) =

Step 8. We will get a contradiction in this step. If ^A(^n(l5 '^)) < C\ — {\ — \ra) for some u ^ Q, then by (4.7), hn{l,u) G JF*(A) and (4.19)

$ A ( ' ? * ( 1 , U))

= $A(r/(2eo, ft„(l,«)) < $A(ft„(l,«)))
CA — (A — A„) for some u e Q, then by step 3 and step 4, ||/i„(l,u)|| < k and sup„gQ $A(/in(l,M)) < Cx+£o- Thus, /i„(l,u) G J^e*oWAssume that $x{ri*{l,u)) > CA - (A - A„). Then for 0 < t < 2so, we have, - (A - A„) 1 on J^*Q(A), we see that ^A(^(2£o,/^n(l,^)))-^A(/^n(l,^))) ^^^° d -^x{r](t,K(l,u)))dt (^l(7?(t,/l,(l, ^))), rA(^(t,/ln(l, ^))))^t 0

< -2£o.

Therefore, by step 4, ^A(^(2£0,/^n(l,^))) 0 for all z, t and Gz{z^ 0 / k l ~^ ^ uniformly in t as z ^ 0. (A3) G{z,t) = \A^{t)z ' z + F{z,t), where Fz{z,t)/\z\ -^ 0 uniformly in t as \z\ -^ 00 and Aoo{t)z - z > fiz - z for some /i > /ii. (A4) ^Gz{z,t)'

z-G{z,t)

>Ofor ah z,t.

4.2. HOMOCLINIC ORBITS OF HAMILTONIAN

SYSTEMS

(A5) There exists S G (0,/io) such that if \Gz{z,t)\

65

> (/io — ^)|^|, then

Let Cz := —Jz — Az and denote the inner product in I/^(R, R^^) by (•, •). Note that a{JA)niIl = 0. Then E := V{\/:\i) (V denotes the domain) is a Hilbert space with inner product (Z,'U)D := {z^v) + (|£|2z, |>^|^'^) ^ind E = i!f2(R^R2^). Moreover, to C there corresponds a bounded selfadjoint operator L : E ^ E such that (I/Z, i;)!) = / {—Jz — Az) ' vdt, R

E = ^ + 0 ^ ~ , where ^ ^ are L-invariant and ( Z + , Z ~ ) D = (z+,z~) = 0 whenever z^ G ^ ^ . Also, {Lz,z)j) is positive definite on E~^ and negative definite on E~. We introduce a new inner product in E by setting {z,v) := {Lz^,v^)j)

- {Lz ,v~ IV'

Then {Lz^z)^^ = | | z + p — ||z |p, where || • || is the norm corresponding to (•,•). It is easy to see from the definitions of /io, /i±i that | z + f >;,l(z+,Z+),

(4.21)

\\z-f>-^_,{z-,Z-)

_ and

\\z\\^ > fio{z,z).

Let {Ej^ : fi G R } be the resolution of identity corresponding to C Then EQ is the orthogonal projector of E onto E~ and Ei^{E) D ^ ~ whenever /i > 0. If /i is as in (A3), then /i > /ii and since /ii is in the spectrum of £, it follows that E^{E) ^ ^ ~ and there exists a ZQ G ^ + , ||zo|| = 1, such that (4.22)

/

{-JZQ

-

AZQ

-

IIZQ)

• zodt = 1 - /i(zo, zo) < 0.

By (A2) this implies that Hz{z,t) = Az + c>(|z|) as z ^ 0, where A is independent oft. In general, one can assume that A = A{t); however, as observed in V. Coti Zelati-I. Ekeland-E. Sere [116], in many cases one can get rid of t-dependence of A by a suitable 1-periodic symplectic change of variables. If this is not possible, then the hypothesis a {J A) H iH = 0 in (Ai) should be replaced by the one that 0 lies in the gap (/i_i,/ii) of the spectrum of C = —J-^ — ^(^5 ^^d i^ (^3) the constant ji should be greater than /ii. Of course, (A5) should be changed accordingly. Note that the spectrum of C is completely continuous. That is Lemma 4.4. Let A{t) G C(R, R^^) he a 1-periodic symmetric matrix-valued function and let C = —J-^ — A{t) : L'^ D H^ ^ L'^ be the corresponding selfadjoint operator. Then the spectrum of C is continuous.

66

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

Proof. Since C is selfadjoint, it has no residual spectrum and the isolated points of (J{JC) are eigenvalues. Therefore, it suffices to show that C has no eigenvalues. Assume that (3 is an eigenvalue with an eigenfunction u G H^. Then (4.23)

-. =

j^A{t)^(3)u.

Let W{t) be the fundamental matrix of (4.23) with 1^(0) = / . By the Floquet theory (cf. P. Kuchment [202]), W{1) = P{t)e^^, where T = InVF(l) and P{t) is a 1-periodic continuous differentiable matrix valued function with a bounded inverse P~^{t). Let v{t) = P~^{t)u{t). Then v{t) -^ 0 and \t\ -^ oo and

With respect to the eigenspaces of T corresponding to the positive, negative and 0 eigenvalues, we may split R^^ as R^^ = M^^M©M^. Assume P*, * = +, —, 0, are the projections from R^^ to M*. Then

Note that v'^{t) ^ 0 as \t\ -^ oo, we must have v'^{t) = 0. This implies i^ = 0, a contradiction. D Theorem 4.5. Assume {Ai)-{A^). clinic orbit.

Then system (H) has at least one homo-

It follows from (^2) and (^3) that \Gz{z,t)\ < c\z\ for some c > 0 and all z^t. Therefore (4.24)

^{z) :=]- I {-Jz 2 JR

- Az) - zdt - [

G{z,t)dt

JR

is continuously differentiable in the Sobolev space H'^iYi^Yi?^)^ and critical points z ^ 0 of ^ correspond to homoclinic solutions of {H). Let '0(z) := / G{z^t)dt. Clearly, V^ > 0, and it follows from Fatou's lemma that V^ is JR

weakly sequentially lower semicontinuous. Since \Gz{z^t)\ < c\z\ and Zn ^ z implies z^ ^ ^ in I/^^^^(R, R^^), it is easily seen that V^^ is weakly sequentially continuous. Thus, (c) of Theorem 4.3 is satisfied. Set

4.2. HOMOCLINIC ORBITS OF HAMILTONIAN SYSTEMS

* A W := l\\z+f

- A ( ^ | | z - f + ^{z)),

67

1 < A < 2.

Then ^ i = ^ (cf. (4.24)). Choose a ZQ G E~^ as mentioned above and let B:={zeE^:\\z\\=r}; M :={z = z- ^ pzo : \\z\\ 0},

A = dM,

where R > r > 0 are to be determined. Lemma 4.6. There exist r > 0 and b > 0 such that ^X\B ^ b. Proof. Choose p > 2. By (^2) and (A3), for any s > 0 there exists a C^ > 0 such that G{z,t) < s\z\'^ ^ CS\Z\P. Hence ^{z) = [ G{z,t)dt < e\\z\\l + CM\l

< c{e\\zf +

CMH-

It follows that '0(^) = o(||zp) as z -^ 0 and there are r > 0, 6 > 0 such that ^x{z)>b>OfoT

z e B.

D

Lemma 4.7. There exists an R > r such that ^x\dM ^ 0Proof. Since G{z,t) > 0 according to (^2), we have ^z-)

1, = --\\z-f-

/I

G{z-,t)dt G{z-,t)dt 0, a contradiction. Consequently, there exists an i? > 0 such that ^ A ( ^ ) < ^(^) < 0 for z G dM. D

Combining Lemmas 4.6, 4.7 and Theorem 4.3 we obtain Lemma 4.8. For almost every A G [1,2] ^/lere exists a bounded sequence (zn) C E such that ^'^{zn) -^ 0 and ^x{zn) -^ c\. Let {zn) C ^ be a bounded sequence. Then, up to a subsequence, eiry+R

ther (i) Vanishing: Nonvanishing:

lim sup /

|z-^p(it = 0 for all 0 < i? < oo, or (ii)

there exist o^ > 0, i? > 0 and T/^ G R such that ry^+R

lim /

|z^p(it > Q^ > 0.

Lemma 4.9. For any hounded vanishing sequence {z^) C E, we have lim / G{zn,t)dt=

lim / G^{zn,t) - z^dt = {).

Proof. Recall the concentration-compactness Lemma L17 due to P. L. Lions [228]. Although this lemma is stated for z G H^ ^ by a simple modification, the

4.2. HOMOCLINIC ORBITS OF HAMILTONIAN

SYSTEMS

69

conclusion remains valid in H^. Therefore, if {z^} is vanishing, then z^ ^ 0 in L^ for all 2 < 5 < oo. On the other hand, by assumptions (^42) and (A3), for any s > 0 there exists a C^ > 0 such that (4.26)

\G,{z,t)\<s\z\^C,\zr\

where p > 2. Hence, / G(zn, t)(it < c(£||zn|p + C^Hz^H^) and

// R.

| G , ( z „ , t ) | \zt\dt

< C{S\\Z„\\ \\zt\\

+ Cell^nlirl^lW,

and the conclusion follows.

D

L e m m a 4.10. Let A G [1,2] be fixed. If a bounded sequence {vn} C E satisfies 0 < lim^^oo ^A('^n) ^ cx ^^^ l™n^oo ^A('^^) ~ ^' then there exists a yn ^ Z such that, up to a subsequence, Un{t) := Vn{t -\- y^) satisfies Un^ux^O, ^x{ux)

>

l i m {^x{Un)

-

-(^l(^n),^n))

X f {lG,{ux,t)-ux-G{ux,t))dt R 2

=$ A K ) . D

70

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

Lemma 4.11. There exists a sequence {A^} C [1,2] and {zn} C ^ \ { 0 } such that An ^ 1, ^A^ (zn) < cx^ and ^'^^ (zn) = 0. Proof. This is a straightforward consequence of Lemmas 4.8-4.10.

D

Lemma 4.12. The sequence {z^} obtained in Lemma 4-11 ^s bounded. Proof. Assume \\zn\\ -^ oo and set w^ = ^n/||^n||- Then we can assume, up to a subsequence, that Wn ^ w. We shah show that {wn} is neither vanishing nor nonvanishing, thereby obtaining a contradiction. (a) Nonvanishing of {wn} is impossible. If {wn} is nonvanishing, we proceed as in the proof of Lemma 4.10 to find a > 0, R > 0 and y^ ^ Z such that if Wn{t) := Wn{t -\-i/n), then / \wn{t)fdt > a for almost all n. J-2R Moreover, since ^^ {z^) = ^^ {z^) = 0, where Zn{t) = Zn{t -\- y^), for any ( / ) G C ^ ( R , R 2 ^ ) wehave (4.29)

{w:^,(l)) -Xn{w-,(l))

-Xn /

A^{t)wn'(l)dt

Jn

-An / p^ \Wn\dt = {). Jn \Zn\ Since ||w;^|| = \\wn\\ = 1, Wn ^ w in E, w^ ^ w in Lf^^CR^R?^) and '^n(0 ~^ ^if) ^'^' i^ ^ - I^ particular, w; ^ 0. Since \Fz{z^t)\ < c\z\ for all z,t, by {H^) and Lebesgue's dominated convergence theorem and by passing to the limit in (4.29), it gives (w;^, (j)) — {w~,(j)) — / AoQ{t)w - (j)dt = 0, that is, equation z = J{A-\-Aoo{t))z has a nontrivial solution in E, which contradicts Lemma 4.4. Therefore nonvanishing of {wn} is impossible. (b) Vanishing of {wn} is impossible. Suppose that {wn} is vanishing. Since ^A i^ri) = 0, we have

(^1JZ,),4) = | | 4 f - A, / G.(^n,t) . Z^dt = 0, Jn

{^'^Jzn),z-)

= -Xn\\z-f

- Xn / G,{zn,t)'Z-dt Jn

Since ||i(;nll^ = \\w:^P + ll'^n IP = 1^ we have that

n

G^(z^,t) • (A^^+ -w~)dt=

\\zn\\.

= 0.

4.2. HOMOCLINIC ORBITS OF HAMILTONIAN

SYSTEMS

71

\G (z t)\ Let Sji := {t G R : —^ — < /j^o — S}. By using Holder's inequality, the relation {w~^,w~) = 0 and (4.21), we see that

e.

IknII

< (/io -^)An||^n||2 ^ (/JQ - 5)\n

< 1 for almost all n. Hence (4.30)

hm /

^.(^n,t).(A^+-^-)^^^^^

and since \Gz{z,t)\ < c|z|, it follows that

R\e^

IknII

/ > /

' Zn

-G{Zn,t))dt

{]-G,{Zn,t)-Zn-G{Zn,t))dt 5dt

-^ 00.

However, recalling that ^A^(^n) ^ c^^ and (^^ {zn),^^) = 0, we obtain 1 {-Gz{Zn,t) R ^

a contradiction.

cx • Zn -G{Zn,t))dt

< —^ < OO, A^

D

72

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

Proof of Theorem 4.5. We have proved that there exist A^ ^ 1 and a bounded sequence {zn} such that ^A^(^n) < CA^ and ^^ (zn) = 0. Therefore

= $ ^ j 2 „ ) + ( A „ - l ) ( z - + t/.'(2„)) = (A„-l)(z-+^'(z„))

Since ($^ (^^n), ^^) = 0, by (4.26), we obtain that (4.31)

||z+f = \nj^G,{Zn,t)

• Z+dt < \\\Zn\? + C||z„r,

(4.32)

\\z-f = -j^G,{zn,t)-z-dt 2. Hence

\Znf c for some c > 0. If {^n} is vanishing, it fohows from Lemma 4.9 that the middle terms in (4.31)-(4.31) tend to 0; therefore z^ -^ 0. Hence {zn} is nonvanishing. Note that if the sequence {v^} in Lemma 4.10 is nonvanishing, then the hypothesis hm ^x{vn) > 0 may be omitted. Therefore, there exist y^ ^'Zisuch that if Zn{t) := Zn{t -\- yn), then z^ ^ z ^ 0 and D ^\z) = 0. This completes the proof. Notes and Comments. Some authors studied homoclinic orbits for Hamiltonian systems via the critical point theory. The second order systems were considered in A. Ambrosetti-M. L. Bertotti [14], A. Ambrosetti-V. Coti Zelati [16], V. Benci-F. Giannoni [54], P. Caldiroli-L. Jeanjean [76], P. C. CarriaoO. H. Miyagaki [79], Y. Ding [132], Y. Ding-M. Girardi [133], F. Giannoni-L. Jeanjean-K. Tanaka [173], W.Omana-M. Willem [268], E. Paturel [274], P. Rabinowitz [294], P.Rabinowitz-K.Tanaka [297] and V. Coti Zelati-P.Rabinowitz [117]; and those of first order in G. Arioli-A. Szulkin [22], V. Coti ZelatiLEkeland-E. Sere [116], Y. Ding-S. Li [134], Y. Ding- M. Willem [136], H.HoferK. Wysocki[188], E. Sere [334, 335], K. Tanaka [363], C. A. Stuart [353] and A. Abbondandolo-J. Molina [3]. Basically, in all these papers the nonlinear term was assumed to be superlinear. Lemma 4.4 is due to Y. Ding-M. Willem [136]. Theorem 4.5 was obtain by A. Szulkin-W. Zou in [361]. Some Poincare-Melnikov type results for Homoclinics can be seen in A. AmbrosettiM. Badiale [12]. See also the survey in T. Bartsch-A. Szulkin [40].

4.3. ASYMPTOTICALLY

4.3

LINEAR SCHRODINGER

EQUATIONS

73

Asymptotically Linear Schrodinger Equations

Consider the Schrodinger equation (SEi)

- A ^ + V{x)u = / ( x , u),

where x e K^, V e C ( R ^ , R ) and / G C(R^ x R , R ) . Suppose that 0 is not in the spectrum of - A + V in ^ ^ ( R ^ ) (denoted by 0 ^ cr(-A + V)). Let /ii be the smallest positive and /i_i the largest negative /i such that 0 G cr(—A -\- V — fi) and set /io •= min{/ii, —/i_i}. It is well-known that if V is periodic in each of the x-variables, then the spectrum of —A + V (in I/^) is bounded below but not above and consists of disjoint closed intervals (see M. Reed-B. Simon [301, Theorem XIII. 100]). Similarly, we introduce the following hypotheses. (Bi) V is 1-periodic in Xj for j = 1 , . . . , N, and 0 ^ cr(—A + V). (B2) / is 1-periodic in Xj for j = 1,...,A^, F{x^u) / ( x , u)/u -^ 0 uniformly in x as i^ ^ 0. (B3) f{x,u) = Voo(x)u -\- g{x,u), where g{x,u)/u |i^| -^ 00 and 1^00(^) ^ /^ for some /i > /ii.

> 0 for all x^u and -^ 0 uniformly in x as

(B4) ^uf{x, u) — F(x, 1^) > 0 for all x, i^. (B5) There exists a (5 G (0,/io) such that if/(x,i^)/i^ >/io—^, then ^i^/(x,i^) — F{x,u) > S. Theorem 4.13. If the hypotheses (Bi) — (B^) are satisfied, then (SEi) has a solution u ^ 0 such that u{x) ^ 0 as \x\ ^ 0 0 . The functional ^{u) :=l

[

{\Vuf + V{x)u^)dx - [

F{x, u)dx

is of class C^ in the Sobolev space E := H^{Rj^)^ and critical points of ^ correspond to solutions u of (SEi) such that u{x) ^ 0 as |x| ^ 00. If a{—A-\V) n (-00,0) ^ 0, then E = E^ QE-, where E^ are infinite-dimensional, and the proof of Theorem 4.13 follows by repeating the arguments of previous sections. Notes and Comments. If cr(—A+ y ) C (0, 00), then E~ = {0}, /i_i = —00 and ^ has the mountain pass geometry. Theorem 4.13 remains valid in this case, and it is in fact already contained in Theorem 1.2 of L. Jeanjean [191],

74

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

where V = constant. For general asymptotically linear cases with periodic potentials, the first result (Theorem 4.13) was due to A.Szulkin-W. Zou in [361] (and later in G. Li-A. Szulkin [209]). For the asymptotically linear cases of non-periodic potential, see L. Jeanjean-K. Tanaka [195] (where V{x) -^ y(oo) > 0 as \x\ -^ oo), C. A. Stuart-H. S. Zhou [354] (where the problem has radial symmetry) and also F. A. van Heerden [370], F. A. van Heerden-Z. Q. Wang [371]. The superlinear case for (SEi) was studied in S. Alama-Y. Y. Li [6], V. Coti Zelati-P.H. Rabinowitz [118], Y. Ding-S. Luan [135], W. Kryszewski-A. Szulkin [201], C. Troestler-M. Willem [372] and the survey by T. Bartsch-Z. Q. Wang-M. Willem [45]. The variational perturbative methods and bifurcation from the essential spectrum can be seen in A. Ambrosetti-M. Badiale [13]. The existence result of a ground state for nonlinear scalar field equations had been obtained in H. Berestycki-P. L. Lions [58]. The readers will be seeing more notes and comments following the next section.

4.4

Schrodinger Equations with 0 G Spectrum

Consider a nonlinear Schrodinger equation with periodic potential: . ^

. ^^

J —Au-\-V{x)u = g{x,u) \ u{x) ^ 0

for X G R ^ , as \x\ -^ oo.

We assume that 0 is an end point of the purely continuous spectrum of the Schrodinger operator — A + V. We introduce the following conditions. (Co) V G C(R^, R) is 1-periodic in x^,

z = 1 , . . . , AT.

(Ci) 0 G c r ( - A + y ) , and there exists a / 3 > 0 such that (0,/3]ncr(-A+y) = 0 . (C2) g G C ( R ^ x R , R) is 1-periodic in x^, z = 1 , . . . , A^. There exist constants ci, C2 and 2 < /i < 2* such that ci|i^|^ < g{x^u)u < C2\u\^ for all X G R^,i^ G R. (C3) g{x, u)u - 2G{x, u)>0

for ah x G R ^ , u G R\{0}. Q(X

U}U

(C4) There exists a 70 > 2 such that liminf ——^—- > 70 uniformly for u^o G[x,u) XGR^.

/ ^ \ rr^i 1 1 1 ^ g(x,u)u — 2G(x,u) (C5) There exists a c > 0 such that limmi :—: > c uni^

^

\u\^oo

formly for x G R ^ . Here, a > a'' := /i — 1

|l^|«

— — if 2* < 00; a > 2*/i — 2* — /i

4.4. SCHRODINGER

EQUATIONS

WITH 0 G SPECTRUM

75

Assumption (CQ) implies that the Schrodinger operator S := —A + V (on I/^(R)) has purely continuous spectrum, written a{—A-\-V), which is bounded below and consists of closed disjoint intervals (cf. M. Reed-B. Simon [301, Theorem XIII. 100]). Assumption (Ci) means that V cannot be constant. It is easy to check that the classical Ambrosetti-Rabinowitz superquadraticity condition (see A. Ambrosetti-P. Rabinowitz [19]): (4.33)

37 > 2 such that 0 < jG{x, u) < g{x, u)u,

\/u G R\{0}, x G R ^ ,

implies (C3) - (C5). But the converse proposition is not true. Here we give an example. E x a m p l e 4.14. Let g{x,t)

=

(4.34)

fi\t\^-H + (/i - 2)(/i -

s)\t\^-^-Hsm\^-^)

+(/i-2)|tr-2tsin2(^),

where 2 * > / i > 2 , 0 < £ < min{/i - 2, /i - /i*}. Hence G{x,t) = | t r + (/i - 2)1^1^-^ s i n 2 ( f f ) . s

Then (C2) - (C5) hold with 70 = /i, o^ = /i — £. However, for any 7 > 2, let tn = (£(n7r + f7r))^^^ Then g{x,tn)tn

-jG{x,tn)

= (/i-7)|tnr + ( / i - 2 ) ( / i - e - 7 ) | t n r - ^ s i n 2 ( ^ ) + (/i-2)|t,rsin2(^)

-^ —00

as n ^

00,

i.e., condition (4-33) can not he satisfied for any 7 > 2. T h e o r e m 4.15. Assume (CQ), (Ci)-(C^). Then solution u e i?ioc(R-^) n L^{R^) for id<s'-'^u + f{x,u),

/3o > 0,

^£(2,2*).

76

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

(Di) / G C(R^ X R, R) is 1-periodic in x^, z = 1 , . . . , AT. f{x,u)u

= o(|i^|^)

as |i^| -^ 0,

f{x,u)u

= o{\u\^)

as |i^| ^ oo

uniformly for x G R ^ . (D2) 0 < f{x,u)u

< ^ ^ ^ ^ ~ ^ V r for all xeK^.ue

Theorem 4.16. Assume (Co), (Ci), equation r -/\u^V{x)u \

Let X := H^{Ii^).

(Di) and (D2). Then the Schrodinger

= (3o\u\^-^u^ u{x) ^ 0

has a nontrivial solution u G H'^^^{BJ^)

R\{0}.

f{x,u)

forxeK^, as \x\ ^ 00

n I/^(R^) for fi < s i?) < ^- Then for y G R ^ with \y\ > R-\-2 we have ||i^||L«=(s(y,i)) ^ ce. This implies that u{x) ^ 0 as \x\ ^ 0 0 . D L e m m a 4.18. There exists a c > 0 such that |5^-|^ 0 there exists Cr,£ depending on r and s such that (4.38)

\Un\L^*{B{y,r))
c} with Un —^ u. Write Un = u^ ^ u~ with i^+ G E^^ u~ G E~] by the definition of the | • |^topology, we observe that u^ -^ u^ in ^ ^ and hence sup ||i^^||£; < 00. Note

82

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

that Hx{un) > c and G{x, u^) > 0, so we have sup \\u~ H^; < oo. On the other hand,

< I

G{x,Un)da

4111-olKIII

A V2 < OO.

Thus, sup ||i^nlU < OO. Insert Lemma 4.17 and assume that Un ^^ u in E^^

Un ^ u strongly in I/[^^(R ) and

Un{x) -^ u{x) a.e. X G R ^ . By Fatou's Lemma and the weak semicontinuity of the norm, we see that c < Hx{u), i.e., Hx is | • l^^-upper semicontinuous. Let i^^ ^ i^ in Ej^. Then 1^^ ^ 1^ in I / [ Q ^ ( R ^ ) for 2 < t < 2*. Hence, by (C2), g{x,Un) -^ g{x,u) in Li^(R^)and /

g{x^Un)(j)dx ^

I

g{x^u)(j)dx.

Therefore, H'^{un) • (j) -^ H'^{u) • (j) for every (j) G E^^.

D

Lemma 4.21. Assume (C2). Then there exist b > 0, r > 0 such that Hx{u) > 6 > 0 ,

\/ue

E^with \\u\\^ = r , V A G [1,2].

Proof. This is obvious.

D

Lemma 4.22. Assume (C2)- Then there exist R> r > O^d > 0 such that sup Hx = 0

and

dM

sup Hx < d < 00 M

for all A G [1,2], where M:={u

= y^sz^:yeE-,\\u\\^

0}, ZQ G ^ + , ||zo|U = 1-

Proof. For u = y -\- SZQ, by (C2), Hx{u) 0. Hence Hx{u) 0 independent of A such dM

that supil^A < d < oo.

D

M

Combining Lemmas 4.21-4.22 and Theorem 4.3, we get L e m m a 4.23. Assume (Co), (Ci)-(C2). exists {un} C Efj such that sup||i^nlU < ^ '

^xi^n)

For almost every A G [1,2] there

-^ 0 and Hx{un) -^ cx e [b,d].

Lemma 4.24. Suppose (Co), (Ci)-(C^), A G [1,2]. For a hounded sequence {un} C En satisfying lim Hx{un) G [b^d] and lim H'^{un) = 0, ^/lere exists a ux ^ 0 such that H^-^{ux) = 0, Hx{ux) < d. Proof. Write Un = u^ ^u^ with i^+ G ^ + , i^^ ^ F^ . Since sup | then sup ||i^^||£; < oo. If {u^} is vanishing, that is lim sup then by Lemma 4.19, u^ and Holder's inequality.

/

\u'^\'^dx = 0,

0 < i? < oo.

0 in L^(R^) for 2 < t < 2*. Therefore, by (C2)

g{x,Un)u:^dx\ IM-1U/+ i^^|(ix

< C R^

0 such that lim /

|i^+p(ix > Q^ > 0.

Hence, we may find a ^^ G Z ^ such that lim /

\v:^\'^dx>a>

0,

where '^^(x) := u^{x -\- y^)• By the periodicity, the set {v^ := Un{x -\- yn)} ^^ still bounded and lim Hx{vn) e [b,d],

lim i / ^ K ) = 0.

Since sup ||i;n||^ < oo,

sup \\vn\\E < oo,

we have sup||i;+||£; < oo, sup||i;~||£; < oo. We may assume that v^ ^- u'j^, v~ ^- u^ in E^. Since E^ is compactly embedded in L\^^(BJ^) for 2 < t < 2*, it means that {) c\z^l^^.

Hence,

(4.40)

k„|^ 0 such that (4.41)

g{x, u)u > (70 - So)G{x, u)

for all X G R ^ and |i^| < R^. By (C5), there exists an Ri > RQ such that g{x^u)u — 2G{x,u) > c\u\^ for ah X e K^ and |i^| > Ri. Noting (Cs), we may choose a c > 0 small enough such that (4.42)

g{x, u)u - 2G{x, u) > c|^|^

for all X G R ^ and |i^| > RQ. Since Hx^{zn) < d and H'^^ {zn) = 0, we have that

(J--^)(ll4ll|-An||^-|||) 1

70-^0 +An

Ro

< c\Zn\f,

a > 0,

^ ^ ^ J5(0,2i?)

where cj+(x) := z+(x + y^). Set u;-{x) := z-{x + y^), 6J~(x). Then sup p n | U < oo. Assume that (4.43)

uo^ -^ cj^,

uo~ -^ uj~^

Zn -^ ^'^ + ^~ '= z* in ^ ^ .

Note that Ej^ is imbedded compactly in Lf^^(Il^). Z * ,6J, .,+ cj^ strongly in I/J^Q^(R^). Hence, /

\uj~^\'^dx > a > 0,

Zn{x) := cj+(x) +

Thus we have z^ -^

Zn{x) -^ z^{x) a.e. x G R

and it follows that z* ^ 0. Furthermore, since ||i^||^ < 1, we have \\U\\E < 1 for ah u G ^ ^ . Thus, for any h e E* (the dual space of E), h \E^^ ^^, the dual space of Ej^. Hence, 6J+ ^ 6J+, uj~ -^ uj~ in E. By combining H'^^{zn) = 0 and Lebesgue's Theorem, -i7^(z*).(^ = H'^Sz^)^c^-H'{z^)^c^

Xn{g{x,Zn) -

for any

g{x,z*))(i)dx

cf) G C^(R^), i.e., H^z"") = 0 .

Proof of Theorem 4.16. We merely check that ^(x, i^) = (3\u\^~'^u^ satisfies the assumptions of the theorem with ^^ = JJL = a.

D f{x^u) D

Notes and Comments. The equation —Ai^ + V{x)u = / ( x , u) + Ai^, where V is periodic and A lies in the gap of the spectrum (j{S) of S := —A + V^ was first discussed in H. P. Heinz [180, 181] where a nontrivial solution was obtained by using the linking theorem in V. Benci-P. H. Rabinowitz [55]. In particular, he showed that bifurcation may occur. Heinz's approach was subsequently refined in H. P. Heinz-T. Kiipper-C. A. Stuart [182], H. P. HeinzC. A. Stuart [183] and B. Buffoni-L. Jeanjean [73] (and L. Jeanjean [192]). In B. Buffoni-L. Jeanjean-C.A. Stuart [74], they developed an alternative approach which may eliminate the compactness condition. The first result

4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS

89

assuming that zero is an end point of the spectrum is due to T. BartschY. Ding [34] where the Ambrosetti-Rabinowitz superquadratic condition is essential to their arguments. In B. Buffoni-L. Jeanjean-C. A. Stuart [74] it was assumed that min cr(6') > 0. In A. Alama-Y. Li [5, 6], 0 hes in a gap of the spectrum cr(—A + V) and G{x,u) := J^ g{x,s)ds is strictly convex. Without either the convexity or the compactness condition, C. Troestler-M. Willem [372] got a nontrivial solution by a generalized linking theorem due to H. Hofer-K. Wysocki [188]. W. Kryszewski-A. Szulkin [201] obtained one nontrivial solution by establishing a new degree theory and a new linking theorem. By an approximation technique without the new degree, A. A. Pankov-K. Pfliiger [273] also got a similar result for superlinear cases. In H. Zhou [384], V{x) = constant > 0 and g{x,t) = f{x,t)t with f{x,t) = f{\x\,t). For this case, the working space possesses a compactness of imbedding. In T. Bartsch-A. Pankov-Z. Q. Wang [39], Schrodinger equations with a steep potential well which depends on a parameter were studied (see also F. A. van Heerden [370], F. A. van Heerden-Z. Q. Wang [371]). A. Szulkin-W. Zou [361] were the first to consider the asymptotically linear case (including homoclinic orbits of Hamiltonian systems), when 0 lies in a gap of {—A-\-V). The main results of this section are due to M. Willem-W. Zou [378]. When 0 G cr(—A + V), more problems are still open.

4.5

The Case of Critical Sobolev Exponents

Consider the following Schrodinger equation with critical Sobolev exponent and periodic potential: (SEs)

- A ^ + V{x)u = T{x)\uf-^u

+ g{x,u),

u e ly^'^(R^),

where N > 4; 2* := 2N/{N — 2) is the critical Sobolev exponent and g is of subcritical growth. 0;V,T,g

(Di) y , T G C ( R ^ , R ) , ^ G C(R^ X R,R),A:o := inf^^R^ T(x) > are 1-periodic in Xj for j = 1,..., A^.

(D2) 0 ^ cr(-A + V) and cr(-A -^ V) n (-00, 0) ^ 0, where a denotes the spectrum in I/^(R^). (D3) T(xo) := max T(x) and T(x) — T(xo) = o(\x — XQP) as

X

^

XQ

and

V{xo) < 0. (D4) \g{x,u)\ < co(l + \u\P-^) for ah {x,u) G R ^ x R, where CQ > 0 and p G (2,2*). Further, g{x,u)/\u\'^*~'^ ^ 0 as i^ ^ 0 uniformly for x G R ^ .

90

CHAPTER 4. LINKING AND HOMOCLINIC TYPE SOLUTIONS

(D5) g{x, u)u > 0 for all x G R ^ and u j^ 0. Theorem 4.27. Suppose that (DiJ-fD^) hold. If ^ A A A\

^0

(4.44)

N —2

— >

g(x,u)u

,

rrig

where

rria : =

2

"^

max

—•—-7-—,

a;GR^,iiGR\{0}

\u\^

then equation (SE^) has a solution u j^ 0. In particular, if T{x) = ko > 0, (Ds) can be dropped and the same result holds. Does Theorem 4.27 remain true for A^ = 3? This is still open. It is worth noting that the equation (4.45)

-Au^Xu=\uf-'^u,

A^O,

has only the trivial solution i^ = 0 in VF^'^(R^) (cf. V. Benci-G. Cerami [53]). Under the hypotheses on V, the spectrum of —A -\- V in I/^(R^) is purely continuous and bounded below and is the union of disjoint closed intervals (cf. Theorem XIII. 100 of M. Reed-B. Simon [301]). The following example satisfies the conditions of Theorem 4.27. . . _ J c\u\'^*u g[x,u).- 1 .

Let E := VF^'^(R^). It is well known that there is a one-to-one correspondence between solutions of (SEs) and critical points of the C^{E, R)-functional (4.46)

^{u)

:=

I [ ~ ^ ^

{\Vuf^V{x)u^)dx /

^{x)\u\'^*dx-

JR^

/

G{x,u)dx.

JR^

Let {^(A)}AGR be the spectral family of - A + V in ^^(R^). Let E' ^ ( 0 ) L 2 n ^ a n d ^ + := {id-E{0))L'^nE.

Then the quadratic form /

:=

(|Vi^p+

Vv?)dx is positive definite on E^ and negative definite on E~. We can introduce a new inner product (•, •) in ^ such that the corresponding norm || • || is equivalent to || • ||i^2, the usual norm of VF^'^(R^). Moreover, /

(|Vi^p +

JR^

Vu^)dx = ||i^+f - \\u-f, where u^ G E^. Then the functional (4.46) can be rewritten as (4.47)

^u)

= l\\u+f-l\\u-f-^f I

l

l

T{x)\ufdx-f JYIN

G{x,u)dx. JYIN

4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS

91

In order to use Theorem 4.3, we introduce a family of functionals defined by (4.48) $A(n) = i | | u + f - A ( i | | u - f + l Z

\Z

/

T{x)\ufdx+f

Z J YIN

G{x,u)dx) J YIN

/

for AG [1,2]. Lemma 4.28. ^x is \ - l^-upper semicontinuous. continuous.

^^ is weakly sequentially

Proof. Noting that Un := u~ -\- u^^ -^ u implies that Un ^ u weakly in E and 1^+ -^ u~^ strongly in E, thus we see that the proof is the same as that in the previous section. D Lemma 4.29. Assume that V G I / ^ ( R ^ ) (it need not be periodic). for each /i G R there exists a constant c = c{jii) such that \\u\\g < c{^i)\\u\\2,

Then

\/ueE{fx)L^

where q = 2N/{N — 4) if N > A ( q may he taken arbitrarily large if N = A and q = oo if N < A). Proof. Since {^(A)}AGR is the spectral family of —A -^ V in I/^(R^), we have for a fixed /i G R, that E{jii)L'^ is the subspace of L^ corresponding to A < /i. Note that ( - A + X^)U(^)L2 : E{fi)L^ ^

E{fi)L^

is bounded. Let F be a positively oriented smooth Jordan curve enclosing the spectrum of (—A + V)\E{fx)L^' According to formula (in.6.19) of T. Kato [198] (and J. Chabrowski-A. Szulkin [88, Proposition 2.2]): - — [ {-A^V 27Ti JY

-X)-\dX,

ueE{fi)L'^.

Since V is compact and — A + y — A is invertible for each A G F, we obtain the conclusion by the Sobolev embedding theorem. D

Let S •=

inf

^ ^

be the best Sobolev constant, see Chapter 1. Let ^s{x) :--

(£2 + | x | 2 ) ( ^ - 2 ) / 2 ^

92

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

where N = {N{N - 2))(^-2)/4,£ > 0 and ^ G C ^ ( R ^ , [0,1]) with ^{x) = 1 if \x\ < r/2; ^{x) = 0 if \x\ > r^r sufficiently small. Write 'ds = '^t ~^ '^7 ^^^^ i^t ^ E+,d- e E-. Then by Proposition 1.15,

L e m m a 4.30. Set (4.49)

h{u) := i||^.+ f - i | | « - f " ^ ^ ^ T{x)\uf

dx,

ueE.

Then supii < c :

iV||T||(^-2'/2

/or £ sufficiently small, where Z^ := E~ 0 R'^JProof. We first show that

(4.50)

C, :=

sup

a

/

(|V^p + Vu^)dx < \ 1/2*

T(x)|i^P (ixj

. Note that

R^

l^^{\\/^j\'^V{^jf)dx 0 such that ||^"||2*

0. Then we get (4.50) . If A^ = 4, the proof is similar. If i^ ^ 0, then N/2

(4.55)

(j^N{\^u\^^Vu^)dx max/(ti^) = -^ t>0

-

'

'

/

\ (A^-2)/2

N(j^,T\urdx)

94

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

as long as the denominator is positive. By (4.50) and (4.55), we obtain the conclusion of this lemma. D Lemma 4.31. Suppose that g{x^u)/u ^ 0 as |i^| ^ 0 uniformly for x G R ^ and that g is of subcritical Sobolev exponential growth. If a bounded sequence {wn} C E andXn G [1,2] satisfy Xn -^ A, ^'^^{wn) -^ 0, ^x^{wn) -^ c(A), where 0 < c(X) < cX :=

nrr^^^vTTT, then iwn} is nonvanishinq.

Proof. If {wn} is not nonvanishing, then i(;^ ^ 0 in I/^(R^) for 2 < r < 2* by Lemma LI7. Then, (4.56)

/

g{x,Wn)vndx ^ 0 and /

G{x,Wn)dx ^ 0

whenever {vn} C E is bounded. Hence, (4.57)

^xA^n)

-

-{^xAWn),Wn)

= ^ [

T{x)\w^fdx^o{l)

^ c(A). For any (5 > 0, we choose fi > ||y||oo(l + ^)/^- Write Wn = w:^ -\- w~ G E^ 0 E~, and let w:^ = w;^ + z^, with Wn G E{fi)L'^,Zn G {id — E{fi))L'^, where ( ^ ( A ) ) A G R is the spectral family of —A -\- V in L^. By Lemma 4.29, Wn G E and (4.58)

\\w~\\q < C | | ^ ~ | | 2 < Cll^^ll

and

\\Wn\\q < cWWnh

< c\\Wn\\,

where q = 2N/(N — 4) if A^ > 4 and q may be chosen arbitrarily large if N = 4. Therefore,

= -{^xA^n),W~)

-Xn /

- Xn /

T{x)\Wn\'^*~'^WnW~dx

g{x,Wn)w~dx

/ipn||2- For any

\\V\\^{1^6)/6, [

\Wzn\^dx>6{fi-\\V\U\\zJl>-

Vzldx.

It fohows that, (4.60)

(1-^)/

\Vzn\^dx<j

{\Vzn\^^Vzl)dx.

By (4.59) and (4.60), we have the fohowing estimates: (A /

T{x)\wnfdxf^^*

0 small enough ^

Q

such that, for almost all X G [1,1 + SQ], there exists a ux j^ 0 such that ^ A ( ^ A ) = 0, ^ A ( ^ A ) < supg ^ . Proof. By hypotheses {D4) and (-D5), there exist i? > 0, 5o > 0, such that inf^A>0,

sup^A 0 and Q C Z^, we get that 0 < CA < s u p ^ < sup/i < c*,

(4.61)

where / i , c* and Z^ come from Lemma 4.30. Therefore, there exists SL 60 > 0 such that 0 < CA < c^ for almost all A G [1,1 + So], where c^ comes from Lemma 4.31. For those A, by Lemma 4.31, {un} is nonvanishing; that is, there exist i/n G R ^ , hi > O^Ri > 0 such that limsup /

\un\'^dx > hi > 0. 1)

We may find a T/^ G Z

such that

limsup/ n^oo

Kfd.>.>0,

JB{0,2RI)

where Vn{x) := Un{x-\-yn)- Since V, T and g are periodic, {vn} is still bounded and therefore, lim ^ A K ) e [inf ^A,sup^],

lim ^ A K ) = 0.

4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS

97

We may assume that Vn ^ ux. Since E is embedded compactly in I/[^^(R^) for 2 < t < 2*, we have 0 < K. < lim / ^ ^ ^

\vn\'^dx = I

JB{0,2RI)

\ux\'^dx < \ux\ 2JB{0,2RI)

Hence, ux ^ 0. Since ^^ is weakly sequentially continuous, then ^'^{ux) = 0. By Fatou's Lemma,

=

^A(^A)-2(^'A(^A),^A)

= A /

(^-(T(x)I^Ar + ^ ( ^ , ^ A ) ^ A ) - — T ( x ) | ^ A r

= A /

lim (-{T{x)\vn\'^*


Ri

— ^ -Si

^i(x,z^)z^)(ix

gi{x,Zn)Zn)dx ^ gi{x,

Zn)Zn)dx ^

4.5. THE CASE OF CRITICAL SOBOLEV EXPONENTS Note by (D^) that \g{x,z)z\ < c\zf that

99

for all {x,z) G R ^ x R. Thus we see

ll4f-An|k-f C^

L

C

L L

c^ c c^ c

gi{x,Zn)z

'f2, UjtJ-^

.\>Ri

[i:{x)\Zn .\>Ri

r^

\Zn\^* dx. .\>Ri

However, (4.64)-(4.65) imply that SUp^

>

= >

>

^Aj^n) -

-{^xA^n),Zr,

/

(^-gi{x,Zn)Zn-Gi{x,Zn)jdx

/ J\z^\>Ri

(-gi{x,Zn)Zn-Gi{x,Zn))dx ^

^^

CI

\Zn?* dx.

'\z^\>Ri

It implies that (4.67)

\\z+f-K\\z-fc

A^ /

[i:{x)\Znf

/

\Znfdx.

^g{x,Zn)Znjdx

By (4.67), J^j^ \znfdx < c. Note that {^'y^^{zn), z^) = 0. By {D4) and Holder's inequality, we obtain that

Il4f = A„ /

T{x)\Znf'~'^ZnZ:^dx

0 for ah (t, x, z), z = {u, v) G R ^ ^ .

4.6.1

The Superlinear Case

We need the following assumptions. (El) Wz{t,x,z)

= o{\z\) SiS z ^ 0 uniformly in t and x.

(E2) \Wz{t,x,z)\ < c\z\^ for ah {t,x) and \z\ > Ro, where Ro > O^fi > 0 are constants, 1 < /i < (A^ + 4)/A^. (E3) ^ i y ^ ( t , x , z ) z - i y ( t , x , z ) >c\z\f^ for ah (t,x,z), where

T h e o r e m 4.36. Assume that {Ei)-{Es) hold. Then {SE4) has at least one nontrivial solution. In the next case, the potential satisfies local conditions both at zero and at infinity. (Fi) There exist z/ > p > 2, z/ < {2N + 4)/Ar, ci, C2, C3 > 0 such that c i k r < W.(^,^,^)^ < \W,{t,x,z)\\z\

< C2\z\^ ^ cs\z\P

for ah (t, X, z) G R X R ^ X R ^ ^ . (F2) W^{t,x,z)z-2W{t,x,z)

>0

for ah {t,x,z)

^ (0,0,0).

(F3) There exists a 70 > 2 such that liminf —TTT^—^—^ > 7o \z\^oo

W{t,X,z)

-

^

uniformly for (t, x) G R x R ^ . ^

^ ' ^

4.6. SCHRODINGER

SYSTEMS

103

(F4) There exists an o^ > p such that ^Wz(t,x,z)z — 2W(t,x,z) ^ , . / X -r^ -r^/\r hminf ^ \ , ^ ' ' ^ > c > 0 uniformly for (t,x) G R x R ^ .

Theorem 4.37. Assume that {Fi)-{F4) hold. Then {SE4) has at least one nontrivial solution.

Let

and A := Jo(—A^^ + y ) . Then {SE4) can be rewritten as JdfZ = —Az -\Wz{t^x^z) for z = (u^v). In this way, {SE4) can be regarded as an unbounded infinite dimensional Hamiltonian system in I/^(R^, R ^ ^ ) . Let HQ := L 2 ( R ^ , R 2 ^ ) with the inner product {\J\'^^w, \J\^l^v), Then D{A) = V{JA)

here \J\ =

{-J^f^.

= W^^^ n iyo'^(R^, R^^) and H := L^(R,Ho)

L 2 ( R X R ^ , R 2 ^ ) . Let S = - A ^

=

+ V.

Lemma 4.38. //O ^ cr(-A^ + V), ^/len 0 ^ cr(^) U cr(J'^). Proof. Assume that 0 ^ cr(jr.4). Then there exists z^ = {un^Vn) G ^(^^^4) such that Iknll^ = ll^nll^ + Ibnll^ = 1,

\\JAZr,\\l

= \\Sujl

+ \\Svjl

^

0.

Then, we may assume that ||i^n||2 > c > 0. Let Un := -r.—^. Then ||6'iZn||2 -^ O5 ||'^n||2 = 1- It follows that 0 G cr(6'). This is a contradiction. Similarly, we have 0 ^ cr(^). D Let L = J^t + v4 be the selfadjoint operator on H with domain

V{L) = {ze and norm

iy^'2(R,^o) : z{t) G P(A), / ||^z(t)|||^(it < 00}

104

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

By Lemma 4.38, 0 ^ cr{JA). Then there are r < 0, p > 0 such that (r, p) H crisJA) = 0. Sphtting according to the positive and negative spectrum, we have Let P^ : HQ -^ HQ be the orthogonal projections and {^(A) : A G R } be the spectral family of J A. Then dE(\),

P+ = /

dE{\).

P- = / J — oc

Jp

Set

OO

J{JA) _ I C/(t)=e*^^-^^ = /

^tA, e'^'dEiX). -oo

Then \\U{t)p-U{s)-^\\n

< e-''(*-'*)

for t > s;

\\Uit)P+Uis)-'\\n

< e-''^^-*)

for t < s;

where 0 = min{—r, p}. Lemma 4.39. 0 ^ cr{L). Proof. If it were not true, then there would exist a i^^ G T){L) with ||i^n||2 = 1 and ||I/i^^||2 ^ 0 as n ^ oo. Hence, dtUn = JAun — JLu^ and Un{t)

=

-

I J — oo oo

/

U{t)P-U{s)-^JLUn{s)ds

U(t)P"'U{s)-^JLun{s)ds.

Let (f ^ : R ^ R be the characteristic function of RQ and oo

U{t)P^U{s)-^i'^{t / Then

-

s)JLun{s)ds.

-OO

oo J — OO

If we let C^{t) = e=F^*C=^(t), we have

where * denotes convolution. Note that / ^ C^dt = 1/0, by the convolution inequality. Thus

\\ut\\2 a > 0, lB{y^,l)

108

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

where B{y, r) denotes the bah centered at y with radius r. Similar to the proof of Lemma 1.17, we have that z+ ^ 0 in L^(Ri+^) for 2 < t < {2N + A)/N. By (^i) and (^2), for any £ > 0, there exists a C^ > 0 such that Wz{t^ X, Zn)z^dtdx a/2,

where By periodicity, {zn} is also bounded and moreover lim Hx{zn) e [b,di lim H'^izn) = 0. Without loss of generality, we may suppose that z+ -^ w^, z~ -^ w^. The compactness of the embedding of ^ + into L[^^(R^+^) for 2 < t < 2{N^2)/N implies that w1^ ^ 0, and it follows that wx := w^ -\- w^ ^ 0. Evidently, i 7 ; ( ^ A ) = 0 . Finally, Hx{wx) = X

lim

= lim Hx{zn) £0 > 0-

no

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

By standard arguments, there exists a z* = z+ + z~ such that z+ ^ 0 and H\z*) = 0 . D

Proof of Theorem 4.37. Under the hypotheses (Fi)-(F4), the conclusions of Lemmas 4.42-4.44 are stih true. It suffices to show that {wn} in Lemma 4.44 is bounded. Note that (4.74)

l l ^ + f - A ^ l l ^ - f = A^ /

W,{t,x,Wn)wndtdx

>

c\\wnt.

By (F3), there exist RQ > 0, SQ > 0 such that ro — £0 > 2 and (4.75)

Wz{t,x,Wn)wn

> (ro - £o)W{t,x,Wn)

for \wn\ > Ro-

By (F2) and (F4), there exists a c > 0 such that (4.76)

W,{t,X,Wr^)Wr^-2W{t,X,Wr^)>c\wX

for

\Wr^\ < RQ.

Note that H\^{wn) < d, H'^ (wn) = 0, we see that

il--^)i\\y^if-^n\\w-f) +A„ /

(

JRI+N

)(Wz{t,X,Wn)Wn-W{t,X,Wn))dtdx ro - £

V

Ro

>

C

W{t,X,Wn)

•j'\w^\>Ro

\Wr,

J\w

It follows that / \wn\-^ < c. By (Fi) and (F4), it is easily seen that J\w^\>Ro or a>iy> p. either iy>a>p (i) If j9 < Q^ < z/, then / \wn\^ < c and for t small enough, l\w^\ c > 0, uniformly for (t, x) G R x R ^ . \z\^oo

|Z|^

T h e o r e m 4.45. Assume {Gi)-{G^). solution.

Then {SE4) has at least one nontrivial

In order to prove Theorem 4.45, we first check the conditions of Theorem 4.3. L e m m a 4.46. There exist ro > 0, 6 > 0 such that H\\B > h for all A G [1,2], where B = {z : z e ^ + , \\z\\ = ro}. Proof.

Trivial.

D

L e m m a 4.47. There exist ZQ G E~^ with \\zo\\ = 1 and R > r^ such that HX\A < 0, where A = d{z = z' ^ SZQ : z' G E-,\\z\\ 0}. Proof. Since /3o > /ii, we can find a ZQ G ^ ^ \ { 0 } such that the quadratic form corresponding to — A^^ -\-V — (3o is negative on HZQ 0 E~. Hence,

\\zor-f3o

/

^ 0. Setting tn = Sn/\\wn\\,u~ = w~/\\wn\\, we have tn > \\u~ \\. Since t^ + \\u~ p = 1, we may assume that t^ ^ txj > 0 and u~ -^ u~ weakly in E. Write u = wz^ + u~. Since {ZO,U~)L2 = 0, we have w'^ — \\u~ W'^ — Po /

u • udtdx

= w'^ — \\u~ W^ — (3^ I <w'-\\u-f<w^{l-

• /3otx7^ / (3o /

{wzo-\-u~){wzo-\-u~)dtdx z^dtdx — Po

{u

Ydtdx

z^dtdx)

0, and we get a contradiction.

D

Lemma 4.48. There exist Xn G [1,2], Wn G ^ \ { 0 } such that Xn -^ I ^ ^ A ('^^) 0 anti H\^{wn) < d. In particular, the sequence {wn} is bounded. Proof. The proofs of the existence of Wn^Xn are similar to those of the previous section. We now prove that {wn} is bounded. Since H'^ (wn) = 0

114

CHAPTER 4. LINKING AND HOMOCLINIC TYPE

SOLUTIONS

and Hx^{wn) < d, we have that (4.79)

(i_i)(||^+||2_AJ|^-f) +An /

(-Wz{t,X,Wn)Wn-W{t,X,Wn))dtdx

On the other hand, by {G2) — (G^s), we may assume that (4.80)

W^{t,X, z)z > fiW{t,X, z)

(4.81)

n^(t, X, z)z - 2n(t, X, z) > c\z\'^

for \z\ < Ro; for |z| > R^.

Therefore, by (4.79) and (4.80),

< C-\- C

(W{t,X,Wn)

Wz{t,X,Wn)Wn

/ + + // '\w^\Ro

= [

\Wr,\^'-'^^\Wr,\^'dtdx

J\w^\>Ro r

(1-02

< ( /

\wXdtdx)

^J\w^\>Ro

^

r

ILN

(2N + 4)

" ( /

\wn\ ^

^J ^\w^\>Ro

\ (2Ar + 4)

dtdxj ^



0,\\u\\ = R} U

[NHBR],

link each other in the sense of Definition 5.1, where Br := {u e E : \\u\\ < r}. Proof. We first consider the case of dim N < 00 and identify N with some R ^ . We may assume that \\yo\\ = 1- Let Q = {syo ^v:veN,s>0,\\syo^v\\
60(A) since A links B. By (Ai), the map A -^ a{X) is non-decreasing. Hence, a\X) := —-—— exists for almost every A G A. We consider those A where a^(A) dX exists. For a fixed A G A, let A^ G (A, 2A) H A, A^ ^ A as n ^ 00, then there exists n(A) such that (5.10)

a\X) - 1 < ^(^^) - ^(^) < a\X) + 1, A„ — A

for n > n(A).

Step 1. We show that there exist r „ G T,w := w{\) > 0 such that ||r„(s,u)|| < w whenever ^x{rn{s,u)) > a{X) — (A„ — A). Indeed, by the definition of a(A„), there exists a r „ e T such that (5.11)

sup $A(r„(s,M))< sup $A„(r„(s,M)) - o ( A ) - A | o ' ( A ) + 2|.

122

CHAPTER 5. DOUBLE LINKING

THEOREMS

Combining (5.12)-(5.15) with (Ai), we see that there exists a w{X) := w such that ||r^(5,1^)11 < w. Step 2. By the choice of F^ and (5.10), we observe for all {s^u) G [0,1] x A that (5.15)

^x{Tn{s,u))
a(A) - (A^ - A) > a(A) - s, then by Step 1, ||F^(5o,i^o)|| < tx7(A). It follows that F^(5o,i^o) ^ QsW- By the definition of a(A), we see that the case of ^A(rn(5, u)) < a{X) — (A^ — A) for all {s,u) G [0,1] x A cannot occur. Therefore, Q£{X) ^ 0. Next, we show that (5.17)

ini{\\^'xiu)\\:ueQei\)}=0

for £ G (0,

) sufficiently small. If not, then there exists an SQ G

, , ,^ (a(A)-60(A)), (0, ) such that ||*'A(W)||

>3£o

forallueQeo(A).

Take n so large that (a'(A) + 2)(A„ - A) < £0, A„ - A < SQ. Define (5.18)

Q:„(A)

•.= {ueE:

\\u\\ < t u + l , a ( A ) - ( A „ - A ) < <S>x{u) < a(A)+eo}.

Then Q:O(A) C Qe„(A).

Similarly, Q*Q(A) ^ 0. By Lemma 2.1, we may construct a locally Lipschitz continuous map Hx of E such that

5.2. TWIN CRITICAL POINTS (a)

\\nx{u)\\2eo,yu (c)

123

e

QliX);

{^\{u),nx{u))>OyueE.

Consider the Cauchy initial boundary value problem:

dt

-nxm,u)),

^{0,u)=u.

By Theorem 1.36, there exists a unique continuous solution S,{t,u) such that ^\{^{t,u)) is nonincreasing in t. Define f C(2s,w)

0 < s < 1/2,

tis,u):=\ [ C(l,^„(2s-l,^t))

l/2<s 0 , r > 0, we define (5.25)

Q(£,r,A):= {ueE : \\u\\ <w{X)^4^dA,\^x{u)

- a ( A ) | < 3s,d{u,B)

0,'iueE.

126

CHAPTER 5. DOUBLE LINKING

THEOREMS

Define

(5.31)

\\u\\ < tx7 + 2 + (iA, I ^ A H - a{X)\ < 2si,

( ueE: Qi := 2ei; dist(^i(a,u), B) > 3Ti.

(5.36) If (5.34) holds, since

Il?i(a,u)-?i(a', u)\\w + 2 + dA; |$A(a(to,r„(2s-l,u)))-a(A)| >2ei; d i s t ( 6 ( i o , r „ ( 2 s - l , u ) ) , B ) > 3Ti.

Assume that (5.39) holds. If Ci(Ti,r„(2s - l,u)) G B, then 5o(A)

= <
0 so small that C* + d

< (1 - ^)(1 - £i - |l)P/(P-2)( A.)2/(p-2) .^ ^^

For w e M with ||u;|| = 5, where 5^''^ = 2(1 - Si - f^)/(j)Co),

we have

$AH > A/.H-Hhlli-r«^ll^-cr > >

(A-f)||u;f-Co||u;r-Ci* Az /^ - Ci*

for A > l-si. Obviously, ^ A ( ^ ) < ^o for ^Wv eN and A < 1. Let A^i := N® {swo : 5 G R } , where w^ is an eigenfunction corresponding to the eigenvalue A/, \\wo\\ = L For any £ G (1 — £i, 1) and v G A^i, ^A(^)

= < =
0 such that

(5.47)

ll^lloo = m a x | ^ ( x ) | 0, then (5.55)

Vi >

^

w' + (

^

|v|-Apo|w|)|f| > 0 .

If

(5.56)

M^Y^°^H-VoH 0. If IU ' +1(;| < 4, then by condition (-D3), we have that

V^3 >

(r2 + A/)A 2 , (Afe + Po)A 2 1 / , N2 1 ./T*/ N ^ ^ + ^ ^ -2/^o(^ + ^ ) --M [x)

> ^'' + Y - '^%^ + ^^' + ^ f - '^%^ - p,\vw\ - 1M*(X) >

(((^2A + AA - 2po)(A.A + poA - 2po))^^ _ ^^^ |^^| _ 1 ^ . ^ ^ ^ --M%x).

> 2

^ ^

138

CHAPTER 5. DOUBLE LINKING

Choose ll^ll := l/Q ^x{u)

:= ^o, then \\v\\^ < Ci\\v\\ < Ci\\u\\ = 1. By (Ds),

=

X\\vf^X\\wf-2

>

2[h\\vf

>

J A ( 1 - ^)\\vf ^

>

THEOREMS

f

+ h\\wf

:=

eo.

-

Z

Ak -t- PO

^l)\\wf

+

hxi\\w\\l-fF{x,u)dx)

+ I 2i^sdx

Ai

+ h{l

^k

Afc-l(Afc - po)

-

Z

J A ( 1 - ^)\\vf

.

+ hx,\\v\\l

+ h{l

^k

^

F{x,v^w)dx

^l)\\wf

JQ

- I

Ai

Ak

Al

M*{x)dx

JQ

JQ

D P r o o f of T h e o r e m 5.9. By Lemmas 5.11-5.13, we may find Ro > go > 0. Let A = {u = v^syo:ve

N^-us

> 0, ||^|| = Ro} U [N^-i H 5 ^ J ,

B = {ue N^_, : ll^ll = ^o}, where yo G Ek with \\yo\\ = 1. Then (5.60)

" - ^ := ao(A)

. . . ^^2g2 (Afe-po) SUP$A(U)

^ ^o(A) := inf

$A(U),

for A e ( ^^""^'^^,,1]. By Theorem 5.7, for almost all A G [ ^^""^'^'^ 1], 2(Afe+po) 2(Afe+poj there are two different critical points Ux,vx satisfying (5.61)

$1(UA) = 0,

$,(«,) = a ( A ) > 6 o ( A ) > ^ ^ ^ i ^ ^ ^

and (5.62)

^ I K ) = 0,

^x{vx) = b{X) < ao(A) < 0.

This is the first part of Theorem 5.9. For the second conclusion of the theorem, by (5.61), we choose A^ ^ 1 and u^ such that ^x^{un) = a(A^), ^^ (un) = 0, Therefore, «(^n) =

{f{x,Un)Un-2F{x,Un))dx.

Jn

5.4. JUMPING NONLINEARITIES

139

Note that T{s,u) := (1 — s)u G T and that A^B are bounded; by the definition of a(An), a{Xn) is bounded from below and above by two positive constants which are independent of A^. Recah condition {D4). By standard arguments, it is easily seen that {||i^n||} is bounded. This yields a critical point u^ satisfying

Similarly, by (5.62), we get another nontrivial critical point v* of ^1 satisfying ^i(i;*) AQ > 0 and that (6.1)

C^{n) CV:=

V{A^I^) C R'^^^iSl)

for some TTI > 0, where Co^(O) denotes the set of test functions in Vi (i.e., infinitely differentiable functions with compact supports in O) and H'^''^{Q) denotes the Sobolev space. If m is an integer, the norm in H^'^^iQ) is given by

(6.2)

||«|U,2 := I E

ll^''"ll^ '

\|//|<m

Here D^ represents the generic derivative of order |/i| and the norm on the right-hand side of (6.2) is that of 1/^(0). If m is not an integer, there are several ways of defining the space H^''^(Q), all of which are equivalent. We shall not assume that m is an integer. A typical example of an operator A satisfying these hypotheses is a second order elliptic operator with smooth coefficients applied to functions satisfying zero Dirichlet boundary conditions

CHAPTER 6. SUPERLINEAR

142

PROBLEMS

on a smooth bounded domain in R ^ . Only the abstract properties listed above are relevant to our analysis. Let q he di number satisfying (6.3)

2 0 such that 2F{x,t) F{x,t) := jlf{x,s)ds.

< Aot^,

\t\ < 8, x e Q, where

(A4) There is a function W{x) G L^{Q) such that either ,,,, , F(x,t) Wix)
00 as t ^ 00; or Wix)
00 as t ^ —00,

for X e ft. The function W{x) need not be positive. (A5) There are constants /i > 2, C > 0 such that

fiF{x,t)-tf{x,t)

s>0.

By (^i), we see that ||M'U/c||g -^ 0- Then, for a subsequence, Mv^ -^ 0 a.e. But by (Ai) the integrand of (6.15) is majorized by \M-'(^f{x,u + TVk)-f{x,u))\' 0 sufficiently small, there is an £ > 0 such that ^{u)

>£,

||^||D = P.

Proof. Let Ai > AQ be the next point in the spectrum of .4, and let A^o denote the eigenspace of AQ. Choose M = NQ- HV. By assumption (^2), there exists a p > 0 such that \\y\\v S

6.2. PROOFS

147

for some x G Q. Write (6.18)

u = w^y,

w e M, y e No.

Then for those x e ft satisfying (6.17), 5 < \u{x)\ < \w{x)\ + \y{x)\ < \w{x)\ + 5/2. Hence, \y{x)\ < 5/2 < \w{x)\

(6.19) and

\u{x)\

(6.20)

\Mv->^o

[ u^dx-cj {\Mu\'^ J\u\5

>\\u\\l-\4u\\'-c[

^\Mu\)dx

\Mu\^)dx J\u\>5

(6.21)

> \\w\\% - \o\\w\\^ -c

I \Mw\'^dx J2\w\>5

in view of the fact that \\y\%, = Ao||^|p and (6.20) holds. Note that

In

\Mw\Hx{l-^-o{l))\\w\\l,

\\u\\ 0 sufficiently small, t-^ (2F(x, yo + t(/)) - Xo{yo + t(l)f - 2F(x, yo) + Ao?/g) < 0. Letting t ^ 0, we see that (/(x, yo) - \oyo)(t>{x) < 0,

x eVt.

Since yo ^ NQ, it follows that the conclusion (a) of the lemma holds.

D

Proof of Theorem 6.1. We define (6.25)

^{u) = \\u\\l -2

f

F{x,u)dx.

Under our hypotheses. Lemmas 6.1-6.5 apply, and (6.26)

{^\u),v)

= 2{u, v)v - 2(/(.,^),.;),

u.veV.

By Lemma 6.6 we see that there are positive constants e, p such that (6.27)

^u)>s,

\\u\\v = P

unless Au = Xou = / ( x , u ) ,

(6.28)

ueV\{0}

has a solution. This would give a nontrivial solution of (6.10). We may therefore assume that (6.27) holds. Next we note that ^±Reo)/R^

[ {F{x, ±Reo)/R^el}eldx -oo Jn as i? ^ oo. By hypothesis (A4), since eo ^ 0 a.e. Since ^(0) = 0 and (6.27) holds, we can now apply the usual mountain pass theorem to conclude that there is a sequence {uk} C V such that (6.29)

= \\eo\\l-2

^Uk) -^c>e,

^'{uk) ^ 0.

6.2.

PROOFS

149

Then (6.30)

^{uk)

= PI-'^

I F{x, Uk)dx -^ c

and (6.31)

{^\uk),Uk)

= 2pl - 2{f{',Uk),Uk)

= o{pk),

where pk = ||t^fe||D. Assume t h a t pk -^ oo, and let Uk = Uk/pk- Since ||i//c||D = 1, there is a renamed subsequence such t h a t Uk ^ u weakly in P , strongly in Lf^^{n) and a.e. in O. By (6.26)-(6.30), ^loci cy

Let Oi = {x G O : u{x) ^ 0 } ,

Liu

UJJL

7

1.

O2 = O \ O i . Then by hypothesis {A4),

2F{x,Uk)^2

2—^k ^k

^00,

^o

X e

ih-

If Oi has a positive measure, then ^

u1 dx >

u1 dx -\-

^

W{x) dx -^ 00.

Thus, the measure of Oi must be 0, i.e., we must have u = Oa.e. Moreover, / i F ( x , Uk) - Ukfjx,

Uk) ^2 ^

/^

-1

But by hypothesis (E),

^k

%

which implies t h a t (/i/2) — 1 < 0, contrary to assumption. Hence, the pk are bounded. Therefore, there is a subsequence which converges weakly in V to a limit i^. For any compact subset OQ C O , t h e imbedding of HQ^{ft) in I/^(Oo) is compact. Thus, we may find a subsequence which converges to u in I/^(Oo). For a subsequence, Uk ^ u Si. e. in OQ. By taking a set of compact subsets of O which exhaust O, we can find a renamed subsequence which not only converges to u weakly in P , but also strongly in I/^(Oo) for each compact subset QQ of Q and also a.e. in Q. We claim t h a t (6.32)

/ F{x,Uk)dx^ JQ

(6.33)

/ f{x,Uk)vdx^ JQ

(6.34)

/ f{x,Uk)ukdx^ JQ

/

F{x,u)dx;

JQ

/ f{x,u)vdx, JQ

/ JQ

f{x,u)udx.

v ^ V;

150

CHAPTER 6. SUPERLINEAR

PROBLEMS

To see this, let ^r{t) be the continuous function defined by

(6.35)

Ut)

r = < [

t, r,

\t\ < r, t>r,

—r,

t < —r.

By (Ai), for a given £ > 0, chose r so large that (6.36)

\f{x,t)\<eM'^W-\

\t\>r

and that (6.37)

\F{x,t)-F{x,£^r{t))\<eM^\t\\

\t\>r.

Then (6.38)

/ (F(x, i^fe) - F{x, u))dx

(6.39)

= /

(F(x, ^fe) - F{x,

U^k))dx

J\uk\>r

(6.40)

+ / {F{x, ^r{uk)) - F{x, Jn

(6.41)

+/

Uu))dx

{F{x,Cr{u))-F{x,u))dx.

J\uk\>r

By (6.37), the integrals of (6.39) and (6.41) are bounded by £ / iMukl'^dx < £c\\uk\\^ < £c. The integrand of (6.40) is majorized by cilM^riukM" + \M^r{uk)\ + \M^r{u)\^ + \M^r{u)\) < c{M^r^ + Mr), which is in L^. Thus, the integral in (6.40) converges to zero. These arguments imply that (6.32) is true. In a similar way, we may prove (6.33) and (6.34). Now, by (6.33), we readily have

Hence, i^ is a critical point of ^ . Noting that

6.2. PROOFS

151

we see that (6.34) implies that \Wk\\v^

/ Jn

f{x,u)udx=\\u\\jy.

Thus, Uk ^ u in V. We now obtain a weak solution of (6.10) satisfying ^{u) = c> £. Since ^(0) = 0, we see that u ^ 0. This completes the proof. D We postpone the proof of Theorem 6.2 until the next section. To prove Theorem 6.3, we shall need the following lemma. L e m m a 6.7. Under the hypothesis (A^), there is aT ^ 0 such that ^(Teo) < 0. Proof. We can assume that ||eo||D = 1. Thus, ^(Teo) = T^ - 2 / F(x, Teo) dx (l-^)lkfe|||)-2 [

>{l-^)\\wk\\l-2

Q{x,uk)dx

[ W{x)dx.

The only way this would not converge to oo is if ||I(;/C||D is bounded. But then II'^/CIID -^ O5 and \Tk\ -^ 1. Since ||i//c||D = 1, there is a renamed subsequence

CHAPTER 6. SUPERLINEAR

152

PROBLEMS

such that Uk ^ u weakly in P , strongly in I/^^^^(0) and a.e. in O. Since w = 0 and \T\ = 1, we have u{x) = Teo{x) ^ 0 a.e. Hence, \uk{x)\ = pk\uk{x)\ -^ oo a.e. Therefore,

/ Q{x,Uk) dx -^ —oo, showing that ^{uk) -^ oo. This

completes the proof of the lemma.

D

Proof of Theorem 6.3. Let S = infx) ^ . Then we may find a sequence {uk} C V such that ^{uk) -^ S. By Lemma 6.8, we must have ||I^/C||D < C. Hence, there is a renamed subsequence such that Uk ^ u weakly in P , strongly in L'i^^{Q) and a.e. in O. Now,

= \\u\\'^ - 2 /

F{x,u)dx

= hkWv -2((^fe -u),u)v - 2 / F{x,Uk)dx^2 JQ

- \\uk -u\\^ / {F{x,Uk) -

F{x,u))dx

JQ

< ^{uk) - 2{{uk -u),u)v^2

/ (F(x, Uk) - F{x, u))dx. Jn

Similar to (6.32), it is easily seen that / F{x^Uk)dx ^

I

F{x^u)dx.

We therefore have the limit ^(u) < S, by which we conclude that ^\u) = 0 and ^{u) = S. Hence, i^ is a weak solution of (6.4). We see from Lemma 6.7 that 6^ < 0. Since ^(0) = 0, we see that u j^ 0. This completes the proof. D

6.3

The Eigenvalue Problem

Theorem 6.9. Assume that {Ai)-{A4) hold. Then for almost every (3 G (0,1), the equation (6.42) has a nontrivial solution. infinitely many solutions.

Au = (3f{x,u) In particular, the eigenvalue problem (6.42) has

We also make the following conditions. (Ag) There are a (5 > 0 and a A < AQ such that 2F{x,t) < At^,

n.

\t\ < 6, x e

6.3. THE EIGENVALUE

PROBLEM

153

(Aio) Either / F{x, Reo) dx/R^ ^ oo as R ^ oo or /

F{x,-Reo)dx/R'^

oo as R ^ oo.

Theorem 6.10. If we replace hypothesis (As) in Theorem 6.9 with (Ag) and {A^) with (Aio); then (6.42) has a nontrivial solution for almost every

/?e(0,Ao/A). Theorem 6.11. If we replace hypothesis (Ag) in Theorem 6.10 with (All)

F{x,t)/t'^

-^ 0 uniformly as t ^ 0,

then (6.42) has a nontrivial solution for almost every (3 G (0, 00).

We shall also need the following extension of Theorem 6.4. Lemma 6.12. Let 1 < A < B < 00. Under hypotheses {Ai)-{A4), for each tv > 0 sufficiently small (not depending on A), we have (6.43)

^ A ( ^ ) := MMv

- 2 / F{x,u)dx

> (A - 1)/^^

If we replace hypothesis (As) with hypothesis A < B < 00, then we have (6.44)

q>^^u)>lx-^\n\

||^||D = K..

(AQ), assuming 1 < A/AQ
AQ be the next point in the spectrum of A, and let A^o denote the eigenspace of AQ. We take M = NQ- H V. By hypothesis (^2), there is a ^c > 0 such that ||^||D < n ^ \y{^)\ ^ ^/2, V ^ ^o- Assume that u ^ V satisfies (6.45)

II^IID < /^and \u{x)\ > S

for some x G O. We write u = w -\- y, w ^ M, y ^ NQ. Then for those x G O satisfying (6.45) we have that 5 < \u{x)\ < \w{x)\ + \y{x)\ < \w{x)\ + (5/2).

CHAPTER 6. SUPERLINEAR

154

PROBLEMS

Hence, \y{x)\ < S/2 < \w{x)\. It follows that \u{x)\ < 2\w{x)\

(6.46)

for all such x. By assumption (^i), we have that

u^dx -c [ (|M^|^ + M'^luDdx > X\\u\\l -Xo [ J\u\5 > A||^|||,-Ao||^f-c /

{Mul'^dx

f\u\>6

> (A - 1)11^111, + All^lll, - Aoll^f - c /

\Mw\^dx

J2\w\>5

in view of the fact that ||^|||) = Ao||^|p. Thus, by assumption (Ai) again, (6.47)

$^(u)>{X-l)\\y\\l+(^X-^^-c,\\w\\l-'y\w\\l,

We take /^ > 0 to satisfy 1 —

AQ/AI

^x{u) > (A - l)/^2 ^ f x - ^ -

Mr, < n.

> ci/^^~^. This gives cm^-^ -\^l\\\w\\l>{\-

l)n\

where ||I^||D = n. Hence, (6.43) holds. To prove (6.44) under hypothesis (Ag), let T] = A/Ao and A = (?^, 6 ) . Under hypothesis (Ag) we have in place of (6.47) (6.48) for

||I^||D

$,(^.) > (A - ^)\\y\\l + f A - A _ c,\\wr^^\ < t^. We take ^c > 0 to satisfy

T]

\\w\\l

— A/Ai > ci^c^~^. It follows that

^A(^)

> (A - r^)^^ + (^ - ^ - ^ 1 ^ ' " ' - ^ + ^ ) ll^lll)

for

||I^||D

= n. This gives (6.44), and the proof is complete.

D

We now turn to the proofs of Theorems 6.9 and 6.10. We shall prove Theorem 6.10 first by applying Theorem 5.7 and Lemma 6.12.

6.3. THE EIGENVALUE

PROBLEM

155

Proof of Theorem 6.10. We take ^ = P , A = (?^, 6 ) , where ?^ = A/AQ, B > 1 is a finite number, and I{u) = \\u\\l,

J(u) = 2 / F{x, u) dx.

By Proposition 5.6, the sets A± = [0,^6^060], B = {x e V : \\x\\j) = K.} link each other if 6^0 > tv. In our case the condition (Ai) of Theorem 5.7 is satisfied. To verify (As) of Theorems 5.7, we first observe that ^^(i^) = 0 is equivalent to (6.42) with f3 = 1/A. Now by hypothesis (Ag), at least one of the limits holds: ^ 0 0 as 6^0 ^ 00. Hence, for 6^0 sufficiently large, one of the inequalities ^A(±^oeo)/^o' < ©lleolll) - 2 [{F{x,±Soeo)/S^ (A - ?^)/^^, A G A. This shows that condition (^3) holds. We now apply Theorem 5.7 to conclude that for almost all A G A, there exists an Uk{X) G V such that sup ||i^fc(A)|| < 00, ^^(i^fc(A)) -^ 0 k and that ^x{uk{X)) -^ a{X) > 60(A). Once we know that the sequence {uk} is bounded, we can apply an idea similar to that used in the proof of Theorem 6.1 to conclude that there is a solution of ^^x{u) = 0, ^A('^) = «(A). From the definition, we see that a(A) > {\ — r])H?. Hence, the equation ^^(i^) = 0 has a nontrivial solution for almost every A G A. Since B was arbitrary, the result follows. D Proof of Theorem 6.9. It suffices to choose A = Ao and show that condition (744) implies hypothesis (Aio). To see this, we note by hypothesis (^4) and the fact that eo(x) ^ 0 a.e. that / ^ F ( x , ±6^060) (ix _ as 6^0 ^ 00.

r F(x, ±6^060) . 2 . n

Proof of Theorem 6.11. We let £ > 0 be an arbitrary number. By condition (All), there is a (5 > 0 such that F{x,t)/t^ < e for \t\ < 5 and x G O. By Theorem 6.10, equation (6.42) has a nontrivial solution for a.e. (3 G (0, Ao/e). Since £ was arbitrary, the result follows. D

CHAPTER 6. SUPERLINEAR

156 Proof of Theorem 6.2. By A G (1,B), there exists a ux (A — l)hi^. Choose An ^ 1, that ^x^{un) = 0, ^x^{un) may assume that bo{l) > £ >

PROBLEMS

Theorem 6.9, for each arbitrary B > 1 and a.e. such that ^^(I^A) = 0 and ^A('^A) = «(A) > A^ > 1. Then there exists a sequence Un such = a{Xn) > a(l) > &o(l)- By Lemma 6.4, we 0. Therefore,

We claim that {un} is bounded. Indeed, if ||i^n||D -^ oo, let w^ = u^/Wu^WvThen Wn ^ w weakly in P , strongly in L'i^^{ft) and a.e. in O. If i(; ^ 0 in P , then c

>

2F{x,Un) , I —^ ^—^^

Jn \K

Wn\ dx

>

[

^^^^^^f^\w^\'dx-

Jw^O -^

^n

[

W,{x)dx

Jw=0

OO

and we get a contradiction. However, if i(; = 0 in P , we define a constant tn G [0,1] satisfying ^A^(^n'^n) = max ^A^(^'^n)- For any c > 0 and iD^ = cwn, we have J^ i^(^, w^) dx -^ 0. Thus, ^X^itnUn)

> ^X^{cWn)

= C^K - 2 / F{x,Wn)dx

> C^/2

for n sufficiently large. That is, lim ^Xr^{tnUn) = OO,

(^';^ ( t ^ ^ ^ ) , ^n) = 0.

It follows that, ^Xr^{tnUn) {f{x,tnUn)tnUn ij\X^

tfiUfij

- 2F{x,tnUn))

dx

dx

OO.

By hypothesis (AQ), ^A^('^n) =

/ L{x^Un)dx

> / L{x^tnUn)dx

-^ oo.

However, we have the following estimates which contradict the above conclu-

6.3. THE EIGENVALUE

PROBLEM

157

sion:
L Consider a family of C^(^,R)-functionals (7.1)

^x{u):=\{Oxz,z)-^{z),

Ae[l,K]

under the following assumptions: (Ai) There exist two bounded linear and selfadjoint operators 0'^^\0'^'^^ : ^ ^ ^ such that OA = A O ^ ^ ^ - O ^ ^ ) , A G [1,/^], where (O^i)^,^) > 0 for dl\ z e E and either {O'^^^z.z) ^ 00 or |(0^^^z,z) + ^ ( ^ ) | ^ 00 as

(A2) ^' is compact.

160

CHAPTER?.

SYSTEMS

WITH HAMILTONIAN

POTENTIALS

(A3) There exist two linear bounded invertible operators Bi,B2 : E ^ E such that the hnear operator B{X,u) := p-B^^e'^^^B2 : E' ^ E' for ah 6J > 0 and A G [1,/^] is invertible, where P~ is the projection over E-. For each p > 0, set (7.2)

S:={Biz:\\z\\=p,zeE+}.

Choose a fixed (7.3)

ZQ G ^ + \ { 0 }

and define

Q := {B2{TZQ + Z) : 0 < r < CT, ||Z|| < M , Z G

^-}

for cr > p and M > p. By dQ we denote the boundary of Q relative to the subspace B2{E~ 0span{zo}). Define Q :={i9 e C([0, l]x E,E) :i9 satisfies (Bi), (62) and (63)}, where (61) i^{t,z) = exp(^u;i{t,z)Os,)z

^W{t,z),

where 5^ G [l,/^],cj^ : [0,1] x

i=l

E -^ [0, +00) is continuous and maps bounded sets to bounded sets; l y : [0,1] X ^ ^ ^ is compact; 1^(0, z) = 0 for any z e E; W{t, z) = 0 for any {t,z) G [0,1] x dQ. (62) i9{t,z) = z,

\/zedQ,\/te

[0,1].

(63) i?(0,z) = z,VzGQ. We note that 'd := id e 6 . Moreover, f3{t,'d{t,u)) G 6 for each 1^,(3 e S. We also recall the following proposition. Proposition 7.1. Le^ E be a Hilbert space and let P : E ^ E be compact. Then for any £ > 0, there exists a PQ : E ^ E such that PQ is compact, locally Lipschitz continuous and | | P ( ^ ) - P o ( ^ ) | | <s,

\/ueE.

Proof. For any i^ G ^ , set Uu:={veE:

\\v - u\\ < 1, \\P{u) - P{v)\\ < s}.

Then {UU}U€E is an open covering of £^ and then has a locally finite refinement {Vi}. Let Pi{u) = dist(u, E\Vi),

7i(^t) = ^ ^ ^ ^ .

7.1. A LINKING THEOREM

161

Then {jiiu)} is a locally Lipschitz continuous partition of unity (cf. Proposition 1.3). By the construction, for each Vi we have di Ui e E such that {P{u) - P{u,),v)

< s\\vl

\/ueV,,veE;

V, C U^^.

Evidently, -f^{u){P{u) - P{ui),v) < e-i,{u)\\vl

\/u,v e E.

It follows that (P(u) -J2^iiu)Piui),v)

< e\\v\\,

Wu,veE.

i

Define i

Then PQ is a convex combination of compact mappings. One readily checks that Po satisfies all the requirements of this proposition. D Theorem 7.2. Assume that {Ai)-{As) S > g > 0 such that ^ A ( ^ ) ^ ^5 ^A(^)

^

Wz e S

Q^

VZ G

dQ

hold and that there exist constants

uniformly for A G [1, /^], uniformly for A G

[1,K];

then for almost all A G [l^f^], there exists a bounded sequence {zn} such that ^ ^ ( z , ) ^ 0;

^A(^n) ^ ^A := inf sup

^ A ( ^ ( 1 , Z))

> S.

Hence, ^\ has a positive critical value > 5 for almost all A G [I, K]. Proof. We first show that (7.4)

i9{l,Q)nS^9,

Wee.

For any 'd{t, z) = exp C^uji{t, z)Os,)z + W{t, z) G 6 , where (jOi{t^ z) > 0, 5^ G [1, ^i^] for z = 1 , . . . , n-,^, define w{t,z) :=

^Ui{t,z). i=l

162

CHAPTER?.

SYSTEMS

WITH HAMILTONIAN

POTENTIALS

For any fixed (t, z), if w{t, z) = 0, then ^{t, z) = z -^ W{t, z); if w{t, z) ^ 0, we let K{t,z) :=

(^uji{t,z)si)/w{t,z).

Then K{t,z) G [l,n] and '^(t, z) = exp(tx7(t, ^)OA(t,^))z + W{t, z). Define 5 := B{t,s,z)

:= P~5]"^exp(^tx7(t, ^2(5^0 + z))OA(t,52(5zo+^)))^2,

for t G [0,1], 5 > 0, z G E~. By assumption (A3), 5 is invertible for any (t, 5, z) e [0,1] X [0, a]xE~. Consider the map H{t, 5, z) : [0,1] x [0, a]xE~ -^ E defined by H{t, 5, z) := stzQ + B-^p-B:[^W{t,

^2(5^0 + z)).

Then H is compact. Let

B^, := {zeE-:

\\z\\ < M}

and define Kt : [0, cr] x B^ ^ R x ^ as follows: i^t(5, z) := ( t | | 5 f i/i(t, ^2(5^0 + z))\\ + 5(1 - t),

z + y(t, 5, z)) .

To prove (7.4), it suffices to show that the equation Ki{s,z) = (p, 0) has a solution in [0, cr] x B^. Obviously, the operator Kt is a compact perturbation of the identity which has the following properties: Ko{s,z) = {s,z), i.e., KQ = z(i. Moreover, for any (5, z) G ^([0,cr] x 5 ^ ) , ^2(5^0 + ^) ^ dQ and hence iy(t, ^2(5^0 + z)) = 0,

i^(t, ^2(5^0 + z)) = B2{szo + z).

If Kt(5, ^) = (p, 0) for some (5, z) G ^([0, cr] x 5 ^ ) , that is, 5tZo + Z = 0,

t||5i-^52(5Zo + ^)|| + 5(1 -t)

= p,

then we get a contradiction since 0 < p < a. By the properties of the LeraySchauder degree: deg(K,,[0,(7] x 5 ^ , ( p , 0 ) ) = deg(Ko,[0,(7] x 5 ^ , ( p , 0 ) ) = deg(z(i, [0, cr] X 5 ^ , ( p , 0 ) ) = 1.

7.1. A LINKING THEOREM

163

It follows that the equation Ki{s,z) = (p, 0) has a solution in [0,cr] x 5 ^ , which implies (7.4). Obviously, by (7.4), we see that dx > S > 0 uniformly for A G [1,K].

_ ddx Since the mapping X ^ dx is non-decreasing, the derivative d^ : ^ ' dX exists for almost every A G [1,/^]. We just consider those A where (i^ exists. For a fixed A G [1,/^), let A^ G [l,/^],An > A and A^ ^ A as n ^ oo, then there exists n(A) such that (7.5)

^1 - 1 < "^^ ~ t^ < d'x + 1. for ^ > ^(^)An — A We show, for almost all A G [1,/^], that there exist 'dn ^ B,/i:o := /i:o(A) > 0 such that (7.6)

||^n(l,^)|| < ko

whenever

^xiM^.u))

> dx - (A^ - A).

For this, by the definition of dx^, there exists i^n ^ ^ such that (7.7)

SUp^A(^n(l,^)) < S U p ^ A j ^ n ( l , ^ ) ) < ^ A . + ( A n - A ) .

ueQ

ueQ

If ^A('^n(l,'^)) ^ dx — (An — A) for some i^ G Q, by assumption (^i), (7.5) and (7.7), we have that (7.8)

(0(l)i^n(l,^),^n(l,^))l\\'^'x{u)f

for all u G flsoW- Set . . ^ ^ '

(7.21)

J I

ueE:

B:={ueE:

either lli^ll > A:o + 2 or ^ A ( ^ ) < ^A - f or ^x{u) >dx^f

1 / '

\\u\\ < A:o + 1, | ^ A ( ^ ) - ^A| < ^o/4}.

Then B C O^o(^) and A n 5 = 0. Let u;{s) := 1 for 5 G [0,1] and u;{s) := 1/s for 5 > 1. Set dist(i^. A) X{u) := dist{u,B) +dist('U,A)'

7.1. A LINKING THEOREM

165

and consider the vector field (7.22)

Xl{u):=x{uM\\Xx{u)\\)Xl{u).

Note u^ A impfies that u G ^soWthat

By (7.18), (7.19) and (7.22), we conclude

(7.23)

and ||X^(^)|| < 1

{^'^{u),Xl{u))

>0

for ah u e E.

Moreover, for any u e B C ^eoW^ by (7.14), (7.18) and (7.19), we observe that (7.24)

{^\{u),Xl{u))

> min{3£^/4,3£o/5} = Self A.

Let (3x G C([0,1] X E^E) be the unique solution of the initial value problem df3x{t,u) dt

-X^(/3A),

/3A(0,^)=^.

By (7.23), we see that (7.25)

^^A(/3A(t,.)) ^ ot

^^^

^

Moreover, /3A has the following expression: (7.26)

I3x{t,u) = e x p ( ( y " -x{Px{s,u)M\\Xx{l3x{s,u))\\)ds)Ox)u

+ Wx{t,u),

where W\ is a compact map and Wx{t,u) = - j Jo

~e{T)x{lix{T,u))Lo{\\Xx{(3x{T,u))\\)W{f3x{T,u))dT,

e(r) := [exp(^ j ^

-x{Px{s,u)M\\Xx{Px{s,u))\\)ds)Ox\.

Define (7.27)

Plit,u)

:= pxit,Mt,u)),

V i e [0,1],

^ueE.

For any u G dQ, then f3^{t,u) = f3x{t,u) and ^A('^) ^ Q < dx — £o/3. Hence ueA (cf. (7.20)) and X*(^) = 0. Moreover, by (7.25), ^xiMt^u))

< ^ A ( / 3 A ( 0 , ^ ) ) < ^ A ( ^ ) < ^ < ^A - ^o/3.

166

CHAPTER?.

SYSTEMS

WITH HAMILTONIAN

POTENTIALS

Consequently, (3x{t,u) G A,X'^{(3{t,u)) = 0. It follows that (3x{t,u) = u, W\{t^u) = 0 and therefore, that /3^ G 6 . For any u e Q, we consider two cases: If ^A('^n(l,'^)) ^ dx — (A^ — A), then (7.28)

^ A ( / 3 J ( 1 , ^ ) ) < ^xiMhu))

^A - (An - A), then \\M^,u)\\

{Xn - A).

< ko by step 2. By (7.15),

sup ^A(^n(l, ^)) < (iA + (2 + d'x){Xn -X)dx-{\n-X).

For t e [0,1], by (7.25) and (7.29), (7.30)

dx - (A„ - A)

(7.31)

4, we assume that p q q{p + 1) p(g_+l) m a x {'a'- ; -p'; ———-; a{q + l)' ———-} p{p + l)'
0.

That is, ^ is a positive and symmetric selfadjoint operator. Therefore, there exists a sequence of eigenvalues {A^} C R ^ of .4 with eigenfunctions ((/)^, tjjn) G L^{n) X L^{dn) such that 0 < Ai < A2 < • • • < A^ • • • / oo;

(7.38)

cl)neH^{n),

cl)n\dQ=^n;

(/>! > 0 OU O .

From (7.38), we know that {A^} C R ^ and {(l)n,i^n •= ^nldn) ^ 1/^(0) 1/^(^0) are the solutions of the eigenvalue problem -A(j)n

^

+ (/)n = An(/)n

= An(/>n

in O,

on

on.

For 1^ := 2^ ^n{4^ni V^n) ^ 1/^(0) X L'^(OQ) and 5 G (0,1), define the operator n=l

A' :V{A')^L^{n)

xL^{dn)

170

CHAPTER?.

SYSTEMS

WITH HAMILTONIAN

POTENTIALS

with n=l

Let E^ := V{A^)^ which is a Hilbert space with the inner product and norm 1 /2

{Z,W)ES

= {A^z^A^w),

\\Z\\ES = {z,z)^s ,

where (•, •) is the inner product of 1/^(0) x L'^{dft) given by JQ

JdQ

By the results of J. Thayer [369, p. 187] (see also J. F. Bonder-J. P. Pinasco-J. D. Rossi [60, Theorem 2.1], J. L. Lions-E. Magenes [226] and M. E. Taylor [366]):

E^^L^ioni

if.>i, p>i,

l>^^y

Furthermore, the inclusion is compact if the above inequality is strict. Let E := E^ X E^^ where 5 and t come from (7.36)-(7.37). Then ^ is a Hilbert space with norm || • H^; induced by the inner product (7.39)

{{u,v), ((/), V^))^ = {A'u,A'(l)) +

{A'v.A'^).

Moreover, E has a natural orthogonal decomposition E := E^ 0 E~ ^ where ^ + :={{u,A-^A'u) E- := {{u, -A-^A'u)

:ueE'}, : u G E'}.

We introduce the projections P^ : E -^ E^ given by (7.40)

P^{u,v) :=

hu±A-'A^v,v±A-^A'u).

Consider the operator C : E ^ E defined by (7.41)

C{u,v) := {A-'A^v,

A-^A'u).

Write z := (u^v) e E diS z = z~^ -\- z~ with z^ G ^ ^ , then Cz = z~^ — z~. Consider the functional ^ : ^ ^ R defined by (7.42)

^Z):=1{CZ,Z)E-

^

[

Jan

n{x,u,v)

:= \\Z^\\E - \\Z-\\E

- ^ {z).

7.2. HAMILTONIAN

ELLIPTIC SYSTEMS

171

Then, by a standard procedure, ^ is of C^ and ^^ is compact. The derivative of ^ is given by (7.43)

{^\u,v),{cl>,ij))E Jan

Jan

We say that {u, v) e E^ x E^ is an (5, t)-weak solution of (7.34)-(7.35) if {u, v) is a critical point of ^ . Lemma 7.4. If {u,v) is a critical point of^,

then {u,v) G iy^'(^+^)/^(0) x

iy2,(p+i)/p(^)^

Proof. First, j Hu{x,u,v)(t)j Hy{x,u,v)^lj = {) Jan Jan for any ((/>, ip) G E. We choose V^ = 0, (/> G H'^{ft). Then we have (7.44)

{A'u,A^^lj)^{A'(t),A^v)-

(7.45)

(^"(/),^^i;) - / Huix,u, v)(j) = 0, Jan

(7.46)

(^^(/),^M = M^,^) = / (-A(/> + (/>)^ + / ^ ^ . Jn Jn o). By the basic elliptic theory, we have a i(; G iy2,(p+i)/p(0) such that Aw = w, in O;

^ — = 7Yti(x,i^(x),'u(x)) on ^O.

Thus, (7.47)

0

=

{-Aw^w)(l) Jn

i(;(-A(/) + (/)) + / w/ Hu{x,u,v)(j). n Jan ^V Jan By (7.44)-(7.47), we see that {v - w^Acj)) = 0. Hence v = w. Consider $AW

:=

Ai||^+|||-i||^-|||-^^W(x,u,z;) an 2

D

172

CHAPTER?.

SYSTEMS

WITH HAMILTONIAN

POTENTIALS

where (7.48)

Cxz = Xz^

-Z-.

In view of condition (^3), we may find /i, z/ > 1 such that /i ^ u, fi -\- u < min{/iQ^, Z//3}. Define Bi{u,v) = {p^-^u,p^-\),

(7.49)

B2{u,v) =

{a^-\,a^-\),

where p G (0,1) and a > 1 will be determined as needed according to different situations. Then Bi^B2 : E ^ E are linear, bounded and invertible operators. Lemma 7.5. Given any cj > 0, A > 1, the operator 5(A,C) := P-B^'exp{CCx)B2

: E' ^ E'

is invertible. Proof. For z = {zi,Z2) ^ E, we write z = z~ -\- z~^ with z^ G E^. Then by a simple computation, (7.50)

^,^^z.±Ayz,^z.±Ayz.^^

Let Pi =

(A" + ( - l ) " ) z i + (A" - (-l)")yl-M*Z2 ^ , (A" - {-!)'')A-*A"zi

P2 =

+ (A" + (-1)")Z2 ^

.

Then, by (7.48), £^(z) = {pi,P2). Hence,

exp(CA) = ( . . . . ) ^ + ( . . . . ) ^ +(^-M*.2,^-M^.0(^^ - ^ ^ ) . If z = (i^,-^~M^i^) G ^ ~ , where i^ G ^^, we have by (7.49) that B2Z {a^-^u,-a''-^A-^A'u) and B-^eyip{CCx)B2Z := (qj), where a^-i(exp(AC) + exp(-C)) - a--i(exp(AC) - exp(-C)) Q = f/ =

^

^

'^5

and - a - - i ( e x p ( A C )+exp(-C)) +exp(•. _^ -a-^(exp(AC) + a^-^(exp(AC) -exp(-C))^^_,^^,^^ 2p'

7.2. HAMILTONIAN

ELLIPTIC SYSTEMS

173

By (7.40), it is easy to calculate that p-B^^exp{C/:x)B2Z ~ 2V^^^ >0.

^ y ^ '

2

^^^^

^ ^^^^^^

2

)

Therefore, B{XX) is invertible.

D

Lemma 7.6. T/iere exist p G (0,1) and S > 0 such that ^x{z) > S for all z e S and A G [1,2], where S := {Bi{u,v) : \\{U,V)\\E = p, (t^,'^) G ^ + } . Proof. For z = (i^,^~M^i^) G ^ + , write z := Biz = {p^-^u,p^-^A-'A'u)

:= z+ + z - .

Then by (7.39), ||z||| = 2M^^,^^^) = 2||^|||.. By (7.35), C\z = / (A - l)p^'-^u + (1 + \)p''-^u ((A - l)p'^-^ + (1 + V 2 ' 2

\)p^'-^)A-*A'u ).

and it follows that (7.51)

iCxz,z)E = {A'U, A'U) ((1 + A)p^+'^-2 + i ^ ^ (p2(M-l) + p2(.-l)) j ^

By iBi)-iBs) *(2)

(7.52)

and (7.36)-(7.37),
2

+(A-1)((T2''-I+ 0, where O C R ^ is not necessarily bounded, (3o{fio) is the lowest eigenvalue of A{resp. B). Assume that the eigenfunctions of/3o(/io) are not equal to zero a.e. on O. Moreover, we assume that ll^ll, < c\\A^^^u\\2

for ah u G V{A^/^),

\\u\\q < c\\B^^^u\\2

for ah u G V{B^/^),

^.46)

where 2N 2 < ^ < 00,

Let F{x,s,t), fying

f{x,s,t),g{x,s,t)

N

0, \f{x,s,t)\

+ \g{x,s,t)\ < c{\s\ + \t\ + 1),

for all 5, t G R, X G O. We solve Av = - / ( x , V, w),

(Sp)

Bw = (3g{x, v, w).

Theorem 8.5. Assume that ^—^

> (j)-^{x)v~^{x) — (j)-{x)v~{x),

as t ^ +00, y ^ v,

where a^ = max{±a,0}. Moreover, (/)±(x) > ^ -/3o,

2F(x,0,t) Xb{w)-fio\\w\\l-

Therefore, inf ^x>

-

^+

f

W{x)dx.

W{x)dx for ah A G [1, 2]. We claim that JQ

sup

^A -

E-ndBR

—CO as i? ^ oo uniformly for A G [1,2]. Let {v^, 0) be any sequence in E~ such that P1 = a{vk) -^ oo. Then ^x{vk,0)/pi

= -a{vk)-2

/ Jn

F{x,Vk,0)dx/pi,

where v^ = v^/pk- Since a{vk) = 1, there is a renamed subsequence v^ ^ v weakly in E~, strongly in 1/^(0) and a.e. in O such that

^ - 1 = - f m

[ {.{x){v-{x)f)dx + (/^o +

cl^-{x)){v-{x)f)dx

This is less than zero unless /^oH'^lli = 1- Since a{v) < 1, this would mean that V G E{(3o), the eigenspace of/3o. Thus v ^ 0 a.e. by hypothesis. But then the integral cannot vanish since (/)± > ^ —/3o. Hence, limsup^A(05'^)/tt('^) < 0 a(v)^oo

uniformly for A G [1,2].

8.2. ELLIPTIC SYSTEMS

191

Thus, ^x has the hnking structure described in Example 8.3. By Theorem 8.4, ^x has a critical point for almost all A G [1, 2]. D

Proof of Theorem 8.6. It suffices to interchange the roles of E~^ and E in the proof of Theorem 8.5. D

Proof of Theorem 8.7. By (5i), for any £ > 0, there exists a c > 0 such that F{x,0,w) < £\w\'^ -^c\w\^, where q > 2 satisfies (8.47). Therefore, for any u = (0, w) G ^ + , for ||i^|| sufficiently small and all A G [1, 2], we have that

^AH

=

Xb{w)-2

F{x,0,w)dx

>

6H-2^11^11^-2c||^||^

>

c.

Choose Wo j^ 0 such that Bwo = /io'^o- Define A:=d{u

= u~ -^suo : u~ G ^ " , ||i^|| < i?, i? > 0, 5 > 0},

where i^o = (O^'^o)- We want to show that ^ A U ^ ^ ^^^ some i? > 0 for ah A G [1,2]. Note that ^ A ( ' ^ ~ ) < 0 for all u~ e E~. If this were not true, then there would exist a sequence u^ = S^UQ -\- u~ such that ||i^n|| -^ cxo and ^\{un) > 0. Write u^ = {vn.SnW^). Then

\h{SnWo)

- a{Vn) > 2 / F ( X , Vn, SnWo)dx

Since Ili^nP = b(snWo) -\- a(vn), we may assume that

> 0.

.. ^ ..^

-^ s*b(wo).

Then 5* > 0. Note that

2b{wo) — mo / \wo\ dx = 2fio / \wo\ dx — mo / \wo\ dx < 0. JQ

JQ

JQ

Thus, there is a bounded subset OQ of O such that 2b{wo) — m^o / ^ \wo\'^ < 0.

192

CHAPTERS.

LINKING AND ELLIPTIC

SYSTEMS

Then, Xb{snWo) _ a{vn) _ U

S:

11

..r,

II

119

2 II

26(g^^;o) _ a(i;n) _ —


^^))

= ^n^('W^n) -

/ g{x,Vn,Wn)wn

= 0, and we

observe that XnKWn) =

/

g{x,Vn,Wn)w,

^52) + /

(\Vn\^\Wn\y~^\Wn\dx.

Choose Q = g(2-a) | ^ | E ^ < 1. By (8.50) and (8.46), \Wn\' \v^Mw^\>Ro 2(i-e)

^•53)

< f /

\wXdx]

I [

Iw^l'^dx]



bo; JCa* nB^9ifa*

= bo.

9.1. LINKING AND SIGN-CHANGING

SOLUTIONS

197

Proof. For any F G T* we have that F([O,l],A)n5^0, smd BnX

F([O,l],A)n5^0,

F([0,1],A)CX

CS. It fohows that F([0,1], A) n 5 n 5 ^ 0. Thus, sup ^ r([o,i],A)n<s > sup ^ r([o,i],A)nSnB > inf ^ r([o,i],A)n<sn5 > inf ^ r([o,i],A)n5 > inf ^

which implies that a* > b^. Case 1: In this case we assume that a* > h^. We suppose that /C^* n 5 = 0 and derive a contradiction. Note that for any u G P \ { 0 } , the vector —^^(i^) o

points toward the interior of V- If ^ has no critical point on the boundary of o

P \ { 0 } , then KLa* CV - By the (PS) condition, there are SQ > 0? ^o > 0 such that (9-4)

' II J " „ >^ l + ||$'(u)|| - 0 for all u and

($'(u),X^(u)) > ^ for any u G ^-^[a* - £o,a* + £o]\(/Ca05o-

Let e o : = { u e £ ; : | $ ( u ) - a * | < 3£o}, ei:={ueE: | $ ( u ) - a * | < 2£o}, ^, ,^ ._ distg(M, 62) ^^^' '~ disti a* - £0, then u G ^~^[a* - £o,«* +^o]- Note that \\(j{t,u) -U\\
0. Assume that ^{u) > a* — £o- Then u G ^~^[a* — So^a* -\- SQ]. If there exists a sequence {t^} and ^0 ^ ^ such that a{tn,u) -^ ZQ in E, then cr has to travel at least (^o-units of time, and an argument similar to that of (ii) provides the proof. o

If there exist a sequence {tn} and ZQ G P such that a{tn,u) ^ ZQ in E and o

therefore in X, then there exists a IN such that a{tN,u) G P . The remaining situation is when (9.8)

dist^; f cr([0, oo), ^), X:[a* - SQ, «* + ^o]) := ^i > 0.

By the (PS) condition, there exists an e* > 0 such that

for u G ^~^[a* - £ o , « * +£o]\(^[«* - ^ o , « * +^o])5i> a* — SQ for all t. Then by (9.^

Similarly, we suppose that ^{a{t,u)) (9.10)

(7(t,^)G^-^[a*-£o,a*+^o]\(X:[a*-£o,a*+^o])5i.

Therefore, or

(9.11)

^{a{^,u))

/^f^

= ^{u)^

I

(i^(cr(5,^)) < a * - 2 £ o .

200

CHAPTER 9. SIGN-CHANGING

SOLUTIONS

By combining (9.7) and (9.11) for cases (ii)-(iii), we see that for any u G A*,i^ ^ P , there exists a T^ > 0 such that (J(TU,U) G ^«*-^o/2y p g y continuity, there exists an X-neighborhood Uu such that ^(T,,^n)c^"*-^°/^UP. Since A* is compact in X, we get a TQ > 0 such that cr(ro, A*) C ^«*-^o/4y p We define r

a(2ro5,^),

5G[0,^],

r*(5,^) = 0, hio ^ (A/c-i,A/c) are constants. f(x t)t — 2F(x t) (B5) liminf'^^ ' \ ^—^^ > c> 0 uniformly for x e Q; here a G (1,2) is a constant.

By assumption (^2), the point (a, 6) may or may not lie on any curves Cn or C12 and may even lie outside the square (A/_i,A/+i)^ for all / > k. The

204

CHAPTER 9. SIGN-CHANGING

SOLUTIONS

points a or 6 may be situated across multiple eigenvalues Xi {I > k). In particular, we permit a = b = Xi {Wl > k -\- 1). This means that resonance at infinity can occur at any Xi {I > k -\-l). Assumptions (^3) and (^4) contain the case when lim

^— = Xh-i, a resonant case at the origin. Let Ei denote

the eigenspace corresponding to A/(/ > 1) and N^ = EiU - - - U E^. Define (9.18)

^{u) := - / \Vu\'^dx - / F{x,u)dx,

u G

HI{Q).

We have Theorem 9.3. Assume that f{x,t) satisfies (9.17) and that {Bi)-{B^) hold. Then equation (9.16) has a sign-changing solution u^ with ^{u^) > 0.

The next case includes double resonant, oscillating and jumping nonlinearities. (9.19)

n ^

_^ ^^^^^) ^^

^ ^Q

as t ^ ±00,

where A^ < b±{x) < A^+i (A: > 2). Theorem 9.4. Suppose that (5i), (^3), (^4) and (9.19) hold. Assume that (Be) min{6+(x),6_(x)} ^ Afc; (B7) no eigenfunction of —A corresponding to X^ or A^+i is a solution of —Au{x) = b-^{x)u'^{x) — b-{x)u~{x). Then equation (9.16) has a sign-changing solution u* with ^{u*) > 0.

Let E := HQ{Q) be the usual Sobolev space endowed with the inner product and norm {u,v)=

{\/U'\/v)dx,

\\u\\ = (

\\/u\'^dx]

,

u^veE.

Let X := CQ{^) be the usual Banach space which is densely embedded in E. The solutions of (9.16) are associated with the critical points of the C^functional ^{u) = -\\uf 2

-

f F{x,u)dx, Jn

u G Hl{Q).

9.2. FREE JUMPING NONLINEARITIES

205

By the theory of ehiptic equations, )C = {u e E : ^'{u) = 0} C X. positive cones in E and X are given respectively by

The

PE

:= {u e E : u{x) > 0 for a.e. x G 0 }

and P := {u e X : u{x) > 0 for every x G O}. It is well known that PE has an empty interior in E and P has a nonempty o

interior P= {u e X : u{x) > 0 for all x G ^^dyu{x)

< 0 for all x G ^O}, o

where z/ denotes the outer normal. Therefore, P =P UdP. We rewrite the functional ^ as Hu) = ^WuWl - ^ ( i ( C o + l y + F(x,^))(ix, /

\ 1/2

where ||i^||£; := ( /^(|Vi^p + (Co + l)\u\'^)dx] Then the gradient of ^ at i^ is given by ^\u)

, which is equivalent to ||i^||.

= ^ - ( - A + (Co + l ) ) " ' ( / ( x , u ) + (Co + 1)^) :=u-

Ju,

where the operator J : E ^ E is compact and J{X) C X. In particular, by the strong maximum principle, J\x, the restriction of J to X, is strongly o

order preserving; that is, for any u — v G P\{0}, we have Ju — Jv GP . Since / ( x , 0) = 0, the ±P are invariant sets of the negative flow of the vector — ^ ^ It is easy to check that V is an admissible invariant set. Lemma 9.5. Assume that {Bi)-{B^) hold. Then ^ satisfies the (PS) condition. Proof. Let {un} be a (PS) sequence, that is, ^\un) -^ 0,^(i^^) -^ c. By Theorem L41, it suflices to prove that {un} is bounded in E. In fact, by (^5), there exists an i?o > 0 such that -f{x,t)t

— F{x,t) > c\t\^ for all x G O and

\t\ > RQ. Because of (9.17), we may assume that \f{x,t)t\ < d? for x G O and \t\ > RQ. Then, for n sufliciently large, we have the following estimates:

c+ ll^nll

= ( /

> —C-\- C /

+ /

)(-f{x,Un)Un-

\Un\^dx.

F{x,Un))dx

206

CHAPTER 9. SIGN-CHANGING

SOLUTIONS

Choose Co = (2 - a){N + 2)/(2AT + 4 - Na). Then CQ G (0,1) and

/

\Un\'^dx ,\>Ro „

2CQN

(2Ar + 4)

• /

Ro

^

\ 2JV + 4

^

< ( c + c||w„||)^^^^||w„f^». Consequently,

\\Un\\

=

{^'{Un),Un)


/ooiA

(9.21)

(52 + A / )

/ii :=

;j

2 , (Afe + ^ o )

w -\

IO51 +H ^— +h 1051 Xk — 1^0

2

V —

77^

,

N

F{x,v-\-w).

^ ^0

. Let

208

CHAPTER 9. SIGN-CHANGING

SOLUTIONS

If IU ' +1(;| < 5o, then by condition (^4) and the choice of A/, we have that /il

>

(9.22)

W +

V

--K.o{v^w)

.

(52 + A/) - 2/^0

^ ^

i ^ + 4 ^((g2 + Az-2/^o)(Afe-/^o))V^

2 , (Afe + ^ 0 ) - 2/^0 2

I

I

^ -^ol^^l V

> 0. If IU ' +1(;| > 5o, then by (9.20), we get that (9.23)

/il > /i2 + /i3,

where /n o/i\ (9.24)

A/ - 52 2 , (Afe - ^0) 2 /i2 := — z — ^ + ^ ~A ^ ~ n^vw, o 4

(9.25)

/is := — ^ — ^

^

V - (52 - t^o)vw + ^ - ^ .

We claim that /i2 is always greater than or equal to zero. In fact, if K.o\w\ > 0, then (9.26)

M2 > ^^^w^

+ ( ^ ^ | « | - ^oH)\v\ > 0.

Otherwise, by the choice of A/, we have that /A/ —52

(9.27)

4:Hin , 9

Xk —1^0 9

^2 > ( ^ ^ - l - ^ ) « ^ ' + ^ ^ « ' > 0.

Also, we have that /is

. >

(Az + 5 2 ) - 2 5 2 2 I 1/ ^ ^ - \S2 - f^0\{\v\ (Az+52)-252

2

/

-(^2-^0)H^ + ^ ^

(A/ - 552 + 4/^o) 3(52 - no) 2 I

M I , '^I'^O ^ \w\)\v\ ^ ^—

M l

^ 2

^("^2 - ^0)

2 I '^I'^O

^2

Set (9.28)

Qi := {x en:\v^w\
5o}.

1^;!

9.2. FREE JUMPING NONLINEARITIES

209

Since dim.Ni_i < oo, we may find a constant C/_i such that (9.29)

max|i;| < Ci_i\\v\\

foidllv

e Ni_i.

Let (9-30)

So := —^ r T 7 ^ ( l - T^)8(52 - /^0)Cf_/ Xk^

Then (^o > 0. By (9.22)-(9.28), we have / /j^idx

=

/

>

/

/j^idx -\- /

/j^idx

fiidx

JQ2

2

2

JQ2

If meas02 > SQ, then ^ M x > _ -3(52 v - ^ -^ ' ^o)|| - ^ | | ^ ||2 f +, rSl4 ^^,.

(9.31)

If meas02 < (^o, then by (9.29)-(9.30), (9.32)

f fiidx

>

-^^^^^^Cf_i||i;fmeas02

Combining (B4) and (9.20)-(9.27), we have ^(u)

=

l{\\vf+\\wf)-1

>

\\\vf

+ \\\wf

> \{l-^J\\vf

F{x,v + w)dx + \Xk\\v\\l + \xi\\w\\l -

+ \{l-'p\wf

+ J^f^,dx

>

l n i i n { ( l - g ) , ( l - g ) } | H | 2 + ^^;.idx

>

\{l-j^)\\uf+

f ^^ldx.

jj{x,u)da

210

CHAPTER 9. SIGN-CHANGING

SOLUTIONS

By (9.31), if meas O2 > (^o, then (9.34)

^u)

(9.35)

>

i ( l - g ) | H | 2 - f c - ^ | H | 2 + f^5o

>

i ( l - g ) | H | 2 - f c - ^ | H | 2 + f^ \{1 - g ) | | ^ . f - A ( i _ g ) | | , | | 2 > _L(i _ g ) | | „ | | 2 .

By (9.34)-(9.36), we may find po > 0 and CQ > 0 such that ^{u) > CQ for u G Ni^_^ with ll^ll = po. • Proof of T h e o r e m 9.3. Invoking condition (^3), we readily have ^(u) < 0 for all u G Nk-i. By Lemmas 9.5-9.7, there exist RQ > po > ^ such that ao := sup^(i^) < 0 < Co < 60 •= inf ^(i^), A

B

where A:={u = v^syQ:ve Nk-i.s B:={ue N^_, : ||^|| = po}

> 0, ||^|| = Ro} U [A^^^-i n BR,],

and yo G Ek satisfying \\yo\\ = 1- Theorem 9.2 implies that there is a critical point i^* satisfying ^^(i^*) = 0, ^(i^*) = a* > 60 > 0- Obviously, 1^* ^ 0 and either i^* G 5 or i^* G 5 . The second alternative occurs when ^{u*) = bo := inf^ ^{u). Both cases imply that u* is sign-changing. D L e m m a 9.8. Under the hypotheses of Theorem 9.4, ^{u) -^ —00 for u e N^ as \\u\\ -^ 00. Proof. Note that Hu) = ^\\uf-

f {^h^{x){u^f

^h_{x){u-f

^ H{x,u))dx,

ueE,

where H (x^u) := /

h{x,t)dt;

h{x,t) = f{x,t)

— (b-^{x)t~^ — b-{x)t

j.

Note that min{6+(x), 6_(x)} > and ^ A^, and recall the variational characterization of eigenvalues {A^}. We then have the following estimates for any

9.2.

FREE

JUMPING

NONLINEARITIES

211

ueNk.

= lhf-

[ H{x,u)dx

-\{\

+[

'^^Jb-{x)>b+(x)

-\W--^\ (b-{x)

lb-^{x) — b-{x)]{u'^)'^dx ^

— / Jn

Jb_(x)b+ (x) ^

- /

2

^

b+ix)u'dx

1 2

)(b+{x){u+f + b.{x){u-f)dx

Jb-(x) 0 and q{x) = b-{x) when i(;(x) < 0. Then we have t h a t —Aw = q{x)w. Hence

Ih+f- - I k - f

= / q{x){w^f Jn

- / q{x){w_) Jn

It follows that 0




-Q^(r(^^-))-

It follows that (f G Duj riSn- By the definition of hi{u), ^{^j) ^ ^- Therefore, w{uj) e E^i^uj) ^ En and V(^) G ^n- This implies that V(M n E^) C ^nTherefore, V satisfies all conditions in Definitions 10.6 and 10.7. D Definition 10.9. Let M be an isolated compact subset of KL. A pair (B, B~) of closed subsets of E is said to be an admissible pair for I and M with respect

to 8 if (1) 6 is bounded away from JC\M, 6 ~ C dQ and M C int{Q); (2) I\Q is bounded; (3) there is a neighborhood N of S such that there is an 8-related gradientlike vector field V (called admissible field) for I on N\M; (4) S~ is the union of finitely many closed sets, each of which lies on a C^-manifolds of codimension 1; (5) V is transversal to each of these manifolds at points of Q~ ; (6) the flow T] of —V can leave 6 only via 6 ~ and if u e 6 ~ , then r]{t,u) will leave Q, i.e., r]{t,u) ^ 6 for any t > 0.

Lemma 10.10. Assume I G C^(^,R). Let a < 6, B := /~^([a,6]) and 6 ~ := I~^{a). If 6 is bounded away from JC\{JC H int(6)), then ( 6 , 6 ~ ) is an admissible pair for I and /C H int(B).

10.1. THE KRYSZEWSKI-SZULKIN

THEORY

221

Proof. We first note that there is an open neighborhood UofQ such that U is bounded away from JC\JC H int(6). By Lemma 10.8, there exists an 0 whenever u G B~, then (B, B~) is an admissible pair. D Given u ^ E,£ > 0, set B{u, s) := {x e E : \\x - u\\ < e},

B{u, s) := {x e E : \\x - u\\ < e},

S(u, s) := {x e E : \\x - u\\ = s}.

We have Lemma 10.11. Let U be an open neighborhood of the isolated critical point p of I. Then there exists an admissible pair (B, B~) for I and p satisfying ecu,

I\e-

< I{p) := c.

Further, there is anso > 0 such that B{p, SQ) C int{Q). For any u G 6'(p, £o)n I^, there is a t > 0 such that r]{t, u) G B~, where r] is the flow of —V. Proof. Choose 5 > 0 small enough such that B{p, 5) C U. Let V : B{p, S)\{p} E be an S- related gradient-like vector field with function r]. Then p := M{(3{u) : S/2 < \\u - p\\ < 5} > 0. Choose £ e (0, p6/4). Let 61,62 > 0 he such that 62 < 6i/2 < 6/4 and B{p, 61) c{ueE

: \I{u) -c\
c - s}

and B~ := B n /~^(c - s). Then (B, B~) is what we want. Since mf{(3{u) : ^61 < \\u—p\\ < 6} > 0, there exists an £0 > 0 such that if u G S{p,6i) and I{u) < c, then I{a{t,u))) < c — SQ whenever ||cr(t,i^)|| = ^.

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Choose 62 sufficiently small so that I{u) > c — SQ for each u G B{p,S2). Therefore, a{t,u) cannot enter B{p,S2). Since ^{u) = 1 and r]{u) is bounded away from 0 when 62 < \\u — p\\ < S, I{a{to,u)) = c — £ for some to and r]{to,u) = cr{to,u) G 6 ~ . D Lemma 10.12. Assume that I G C^{E,Il) and that p is an isolated critical point of I. Suppose that ( 6 1 , 6 ^ ) and ( 6 2 , 6 ^ ) are two admissible pairs for I and p. Then Proof. Since ( 6 1 , 6 ^ ) and ( 6 2 , 6 ^ ) are two admissible pairs, we have a neighborhood Ui of B^ and a vector field Vi on Ui\{p}, z = 1,2. By Lemma 10.11, there is an admissible pair (B, B~) for / and p such that B C int(Bi) H int(B2). Thus, we just have to show that H^iSi.S^) ^ H^{e,e-). By Lemma 10.11 and its proof, we get a gradient-like vector field {F^jSp) which is admissible for both (B, B~) and (Bi, B^). Since B~ := Bn/~^(c—e) for some small £ > 0, where c = /(p), the flow rj of —F cannot re-enter B after leaving it. Choose (5 > 0 so small that B{p^ 6) C int(B). Let dist(^,5(p,(5/2)) dist(^, B{p, 5/2)) + dist(^, Ui\B{p, 5)) Then ^ : Ui ^ [0,1] is a locally Lipschitz continuous function. Consider the Cauchy problem: ^ ^ ^

= -a 0. Choose T = (sup^^ / — infei I)/PF' G{t,u) eQfoiT t ( i ^ ) . Define a mapping C : [ 0 , r ] x B i -^ Bi by

[

(T{t{u),u),

te[t{u),T].

Since the function t{u) is continuous, we get that ^ is a deformation of the pairs (Bi,Bj") into {Qo,S]^) and C([0,r] X Qo) C Qo,

filtration-preserving

C([0,T] X B r ) C B ^ .

Further, if i : (Qo, B^) -^ (Bi, B^) is the inclusion and (T •= C{T, •), then (T^i

— id

on

(Qo,B^),

i o C,T — id

on(Bi,B^).

It follows that (Qo, B^) and (Bi, B^) are homotopy equivalent by filtrationpreserving homotopies. Hence, we get (10.17). Combining (10.15) and (10.16), we get the conclusion. D We now can introduce the definition of an 0 such that sup\\I\u)-r{u)\\ e

0. Let £ G (0,?^). By shrinking [/, we may assume that s u p | / | < oo,

u

s u p ||/^(l^) - I\u)\\

< £.

u

It follows that

= {I'{u),V{u)) +

(I'{u)-I{u),V{u))

> r]{u) — £

>0. Therefore, p G B{p, 5) and U\B{p, 5) is bounded away from /C(J). Similar to Lemma 10.8, we may construct an a^_i},

z = O, l, 2, . . . , m;

e," := {u G e^_i : I{u) > a^_i}

for i = 1,..., 771. Note that B ^ = B, BQ = B~ and exc

Hl(Qi,Qi_^)

^ Hl(Qi,Q-),

i = l,2,...,m.

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CHAPTER 10. COHOMOLOGY

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If u e S n I~^{ai), we have that {r{u),V{u)) > 0. It is easy to see that (6^, 6~) is an admissible pair for / and the critical points pj satisfying I{pj) = Cj. By Lemma 10.20, m

(10.26)

m

Mi{e,e-) = ^M|(e,,e-) = ^/?|(e,,e-). i=l

i=l

It follows that each pair (B^, 0~_^) and (B^, B~) are S- finite. The exactness of the cohomology sequence of the triple (B^, B^_i, B^_2) implies that (B^, B^_2) is also S- finite. By induction, (B,B~) = (B^,Bo) is also S- finite. If in (10.24), we replace A by B ^ = B, 5 by B^, L) by B^_i, we obtain Ps{t, B, B,) + Ps{t, B„ B,_i) = Ps{t, B, B,_i) + (1 + t)Q{t, B, B„ B,_i). Summing up, we find m

J2 Ps{t, Oi, Oi-i) = Ps{t, e, e-) + (1 + t)Q{t), where Q{t) has coefficients a^ G [Z+] and a^ = [0] for almost all q. Finally, multiplying (10.26) by t^ and summing over q, we have

M£{t,e,e-) m

= ^P^(t,B„B-) m

= ^P^(t,B„B,_i) i=l

= P^(t,B,B-) + (l+t)Q(t). D Theorem 10.22. Assume that I G C^(^,R) satisfies (PS)*. Suppose that a 1 such that if n > UQ, \\PnLu\\ > c\\u\\ holds for allu G 7^(L) nEn. Proof. Assume that for any A: > 1 there are an n/c > A: and Un^ G 'Tl{L) H En^ such that \\Pn^LunA\ < \\unA/^ properness of L implies that which is a contradiction.

implies PnM^^^

-^ 0. The A-

^^ -^ y ^ ^ ( ^ ) ^^^ \\y\\ = 1. But Ly = 0, ll'^nfc II D

Define the IIQn+ii^^ll - A||T,^|| > co||^||/2. Further, recall that 7^(1/) H F^ is orthogonal to W^. Hence, we have (10.31)

M-(Q,+lL|7^(L)^^.+J = M-{{QnLQn^HnAHn)\n{L)nE^^,) = M-(Q,L|7^(L)^^J+M-(A|^J.

On the other hand, A{Ej^) C Ej^ implies that (10.32)

dn+i -dn =

M-{A\E^^^)

-

M-{A\EJ

=

M-{A\wJ-

By (10.31) and (10.32), we see that M~((5^I/|7^(2.)n£;^) — d^ is Si constant for almost all n. It is finite since M~{QnQA\^(^L^f^En) ^ d^ -\- k, dim.Af{A) < oo and B is compact. D We have

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CHAPTER 10. COHOMOLOGY

GROUPS

Theorem 10.27. (Nondegenerate Case) Assume that I G C^(^,R) and that p is an isolated critical point of I satisfying I{u) = I{p) ^ -{L{u - p),u - p) +(/)(^), where L is an invertible A-proper operator and (j)\u) = o(||i^ — j9||) as u ^ p. If Mg (L) is well defined and finite, then [^],

cui,p)

forq =

[0], If\M^{L)\

M^{L),

•

= +00, then Cl{I,p)

otherwise.

= [0], V^.

Proof. Consider the family of functionals Ix{u) := I{p) + ^{L{x-p),x-p)

+ (1 - A)(/)(^),

A G [0, l],ue

E.

Since L is invertible and A-proper, there exist CQ > 0 and no > 1 such that llP^Li^ll > co||i^||,

u G En,n > no.

Choose £ > 0 such that \\ 0 so smah that 5(0, £)n7^(L) C int(Bi), 5 ( 0 , £ ) n M{L) C int(Bo). Let TT : R ^ [0,1] be a Lipschitz continuous function such that 7r(t) = 0 for t < e/2 and 7r(t) = 1 for t > e. Define V3(^):=^(||/||)Vi(/)+^(||e||)P^,V2(e),

10.3. THE SHIFTING THEOREM

239

It suffices to sfiow tfiat V4 is an admissible field in a neighborhood of ^ B . Let ^ 1 , 6 , ^4 be the flows of - V i , -V2, -V4. Then

satisfies the items 4-6 of Definition 10.9. Since / G 7^(i^) H ^ ^ whenever u e En and n > mo, V4 is >7r(||e||)ryi(/) + 7r(||e||)r?2(e)-£™„, where 6^^ ^ 0 as mo -^ 00 and r]i,r]2 are as in Definition 10.6. We may assume the neighborhood U has been chosen in such a way that u ^ U if \\y\\ < £ and ||e|| < £. Taking TTIQ large enough, we see that (V4(i^), Ly^(j)o{x)) is positive and bounded away from 0 on [/. Hence ( 6 , 6 ~ ) is an admissible pair. D Based on Lemmas 10.28, 10.29 and 10.30, we can finish the proof of the following theorem. We will use the notation as usual: (^1,^2) X {Bi,B2) = ( A I X 5 i , (Ai X B2) U {A2 X 5 i ) ) .

Theorem 10.31. (Shifting Theorem) Assume that U is a neighborhood of an isolated critical point p of I ^ C^(t/, R) and that the operator L in (10.33) is A-proper. If Mg (L) is well defined and finite, then

/ / \M^{L)\ = +00, then Cj{I,p)

= [0],

Vg.

Proof. Let mo be as in Lemma 10.30. Then (61 + Os) nE„ = (Oi n E„) + 0 3 .

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CHAPTER 10. COHOMOLOGY

GROUPS

We compute the cohomology of

((Bi n E^) + 63, ((er n E^) + 63) u ((Bi n E^) + e^)). Topologically this is equivalent to

(ein^^,ern^^)x(e3,e^). Let 5 be a closed ball of dimension dim^ := M~{QnL\^(^L^f^E^)- Then (61 H En, Bj" n En) is homotopically equivalent to {B, dB) for almost all n. By the Klinneth formula, we have

i7^+^-((einK,ernK) x (63,63")) ^i7^+^-((5,a5)x (63,63-)) ^ [i:f*(5,a5)0i:f*(63,6^)]^+^-

If Mg{L) is finite, then q-\-dn — dim^ = q — Mg (L) for almost all n. Hence, we get the first conclusion . If \Mg {L)\ = oo , then q -\- dn — dim^ < 0 or q^dndim^ > dimA/'(L) for almost ah n. Then m+d^-^'^^^^Q^^ Q-) = 0. This completes the proof. D We have the following critical groups for local maximum and minimum. Theorem 10.32. Assume that I satisfies the hypotheses of Theorem 10.31 and Mg (L) is finite. (1) If(j){u) > (j){p) = 0 for all ueU,

then

(

[^],

[

[0],

forq =

M^{L),

Clil.p) = (2) If 4>{u) < 4>{p) = 0 for all ueU, ( CI{I,P) = \ [

[^], [0],

otherwise. then

forq =

M^{L)+M°{L),

otherwise.

Proof. (1) According to the positive and negative spectrum, we split 1Z{L) as following: Tl{L) = E~^ 0 E~. Then we may find a constant CQ such that

10.3. THE SHIFTING THEOREM

241

Write fi{x) = fi^{x) + / i - ( x ) e E^ ® E'. Af{L), we get (10.44)

(/)o

=

By (10.35), for all x G 5(0, (^o) H

i(L/i+(x),/i+(x)) + (/)(p + x + /i+(x)) + 2 ^^'^"(^)' '^"(^^^ + (/)(p + X + /i(x)) -(/)(p + x + /i+(x)).

Let (9(t) : = p + x + / i + ( x ) + t / i - ( x ) , t G [0,1]. Then (10.45)

(/)(p + X + fi{x)) - (/)(p + X + /i+(x)) 0

dt

1

((/)'((9(t)),/i-(x))(it 0 1

{cl)\0{t)) - (j)'{p + X + /i(x)), /i-(x))(it 0

+ ((/)^(p + x + /i(x)),/i-(x)) 1

{^\e{t))-^'{p^x^^{x)),^-{x))dt 0

-(L/i-(x)),/i-(x))

(by (10.34)).

Note that (j)"{p) = 0. We choose 6 sufficiently small so that (10.46)

||(/)^(^(t))-(/)^(p + x + /i(x))||

0 such that I{u) < /(O) for ue E- with ||i^|| < p, I{u) > /(O)

for ueE^

with ||^|| < p.

Theorem 10.33. Let I G C^(^,R) have a local linking at 0 and satisfy the (PS)* condition. Assume that I maps hounded sets into hounded sets. If 0 is an isolated critical point of I, /(O) = 0, E^ = {E~ H E^) 0 {E~^ H E^) and dim(^~ n En) = qo -\- dn for almost all n, then C|«(J,0)^[0]. Proof. Suppose that 0 is the only critical point of / in a ball 5(0, p). Let 0 < ri < r < p and let ( 6 , 6 ~ ) , 6 C 5 ( 0 , r ) , be an admissible pair. Particularly, we may assume that (B, B~) has those properties in Lemma 10.11. Define Bi := {r]{t, u) eS

:t>0,ue

S{0, n ) n

E-}.

Since / < 0 for i^ G ^~with ||i^|| < r i , for u G Bi, there exists a unique t{u) such that r]{t{u),u) G B~. According to the Definition 10.9, t{u) depends continuously on u. Define r]{st{u),u),

if u e Bi, s G [0,1],

u,

if 1^ G B~, 5 G [0,1].

r]i{s,u):--

10.4. CRITICAL

GROUPS OF LOCAL

LINKING

243

It is a filtration-preserving strong deformation retraction of Bi U B Therefore, (10.47)

onto B

i7|(B,B-)-i7|(B,BiUB-).

For 1^ G ^ + , let r{u) := minjri, dist(i^, Bi U B )}, B2 := {u- ^u-^ eE-

®E^ : \\u- \\ < r(^+)}.

Hence B2 is open, E^ C B2 and (Bi U B") n B2 = 0. Set Fr := (5(0, r) n E-) © (5(0, r) n E^). Define inclusion mappings i and j as follows: i :

{B{0,ri) n E-, S{0,ri) n E-) -^ ( B , B i U B - ) , j:

(B,BiUB-)^(F„FAB2).

Then (10.48)

i7|(F„FAB2) ^i7|(B,BiUB-) ^i7|(5(0,ri)n^-,5(0,ri)n^-),

where i*,^* are the induced homomorphisms. Then the mapping

2sriu + (1-25)^-+^+, max{||i^~||,r(i^+)}

se [0,1/2],

riu~ + (2-25)^+, max{||i^~||,r(i^+)}

5 G [1/2,1],

is a deformation of (F^, F^\B2) onto (5(0, r i ) n F , 6^(0, ri)nE ). It preserves the filtration since u^ G E^ whenever u'^ -\- u~ G ^n- Moreover, V2\[0,l]x(B(0,ri)nE-

,S(0,ri)nE-)

is a homotopy between 7^2(1, O^OO and the identity on (5(0, r i ) n F ~ , 6^(0, r i ) n F ~ ) . 7^2 is also a homotopy between (j o i) o 7^2(1,-) and the identity on {Fr, Fr\S2)- Hence, the inclusion mapping j o z is a homotopy equivalence by

244

CHAPTER 10. COHOMOLOGY

GROUPS

filtration-preserving homotopies. Thus, z*oji'* is an isomorphism (see (10.48)). Note i7f ( 5 ( 0 , r i ) n ^ - , 5 ( 0 , r i ) n ^ - ) = [ ^ ] , thus we have

n Notes and Comments. The idea of local linking and related results can be found in H. Brezis-L. Nirenberg [70], K. C. Chang [94], J. Liu-S. Li [233], S. Li-M. Willem [219], J. Q. Liu [230], M. Ramos-S. Terracini-C. Troestler [300], and E. A. B. Silva [336, 337]. If dim ^ ~ < oo, the characteristics of the critical groups with applications were given in S. Li-A. Szulkin [215], K. Perera [278] (Homological local linking), [279] (for asymptotically linear elliptic problems at resonance) and also K. Perera [277]. Theorem 10.33 of this section is due to W. Kryszewski-A. Szulkin [200].

10.5

Computations of Cohomology Groups

Let / G C^{E, R) be a strongly indefinite functional which satisfies the {PSy condition with respect to 0 such that aoo(5 -\- t) < c(aoo(5) + «oo(0) ^^^ ^^^ s,t G R+, • there are two constants a > 0, f3 > 0 such that

10.5. COMPUTATIONS (10.49)

a
0, which is independent of n, such that

{Lu^,u^)>K.\\u^f

for

u^eE^.

We write u = u'^ -\-u~ -\-u^ with u"^ G E^^vP G ker(I/) and set:

(10.50) M:={u:ue E^, ||.+ f " ^ l l - - f " ^ ^ ^ S " "^1'

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CHAPTER 10. COHOMOLOGY

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where the parameters r, T are to be determined later. The normal vector on the boundary dM of V is given by no :=no{n,u)

r u^ = u~^ - du~ - 2'^'(ll^^ll)^;^'

where (10,51)

. ^ ^ ,

m - f ^ .

Next, we show that I\En has no critical point outside V for appropriate r and T. Indeed, by {Ai) and for e > 0 small enough, we first observe that (/'(u),no> = {Lu^, u^) — d(Lu~ ,u~) + (J'(u), no) >K\\u+f + dK\\u-f -c{l+a^{\\u\\Mu+\\+d\\u-\\+Ti}'{\\u^)) > K\\u+f + dK\\u-f - c{l + a^iWu^'W) + \\u+\f-'

+

\\u-f-'){\\u+\\+d\\u-\\+Td'{\\u^))

-ceT'\^'(\\u%f-cs-'al{\\u%-c. By (10.51) and the definition of ttoo and a simple calculation, it is easily seen that

for t > 0. Choose r > (10.52)

, then

cer'\^'{\\u^)\^+cs-'al{\\u^) < cer^ (l + |k0||2)4l|k0||^^ + ll^ II ^ + 11" II (1 + ||U0P)4 +ce' - o 2 i ll + II„,0II2 | | w 0 | | 2 IV

-

„ K

2 l + |k0||2 ^ ^ -

n( l +1 II„,0II2\3'^ ||uO||2)3V

IL.OII ||yO||

" " " l l " II''

10.5. COMPUTATIONS

OF COHOMOLOGY

GROUPS

247

Consequently, for sufficiently large T, we have that

(/>),no)>f(||^+f-^|Kf-r^|M)_, (10.53)

>^T-r - 2 >0.

Let —V denote the negative - | | u - f (||i|| + ce + ced) - (ce-^d + c £ - i ) | | u - f (/^-i) -ceT^{\\u^\\) - ce-'r^(^-'\\\u^\\) - ce-'aU\\u^\\) + J{u') - c, which imphes that ||i^~ + i^^|| ^ oo as I{u) -^ —oo. Now we choose a > 0 such that /C = /C(/) C {z G ^ : \I{u)\ < a}. Then the above arguments imply that there exist b > a and Ri > R2 > 0 such that Mi:={ueM:

\\u^^u-\\

> Ri}

cr^nM CM2:={ueM:

| | ^ V ^ " | | > R2}

Obviously, there exists a geometry deformation retraction "^ of M2 onto Mi. By the (PS)* condition, we may assume that JC{I\E^) C M\I~'^[—b,—a]. Thus the flow of —V provides a strong deformation retraction r] of / ~ " H Ai onto I~^ nAi. Then 1!^ * ?^ is a strong deformation retraction of I~^ H Ai onto Ml. Also by the flow of —V, we obtain a strong deformation retraction of / " n Er^ onto ( / - " n ^ ^ ) U Ai . Therefore,

^ m{{r'' n K) u Ai, /-" n K)

{

T^

\i q = dim E~ + dim ker(I/),

0,

otherwise.

Since (/~^([—a, a]),/~^(—a)) is an admissible pair for / and /C(/) and d i m ^ ~ = Mg (L) -\- dn, we see that (10.55) implies Hl{I,JC{I)) - ^,,M-(L) + MO(L)[-^].

Vg e Z.

(2) Assume that (^2^) holds. We consider T:={ueE:

\\u-f

- d\\u+f - T^\\U^)

< T}.

10.5. COMPUTATIONS

OF COHOMOLOGY

GROUPS

249

Then the normal vector on dT is no := no{n,u) = u

— du'^ —-i}W\u^\

u^

2 ^" "^K||-

By a similar argument, there exist r and T such that (rH,no)0

\\uY)-

10.5. COMPUTATIONS

OF COHOMOLOGY

GROUPS

251

for u G Up\{0}. We conclude that Ix has a unique critical point 0 e Up. Furthermore, SUP{|/A('?^)| :U eUp.Xe [0,1]} < oo and A ^ /^ is continuous uniformly for u e Up. By Theorem 10.15, we have that C|(/,0)-C|(/o,0)-C|(/i,0). Let Liu = Lu-\-u'^ — u~ — u^. Then Li is a bounded linear Fredholm operator of index 0. Hence Li is an invertible A-proper operator. Note that L{En) C E^. Thus it is easy to verify that By Theorem 10.27,

This completes the proof of case (1). (2) The proof is analogous to case (1) by setting 1_ _ . 1 h{u) = -{Lu^u) + (1 - X)J{u) + - A ( | | ^ + f + ll^^f - l l ^ - f ) . D Theorem 10.36. Suppose that I G C^(^,R) satisfies the {PSy condition and that J maps bounded sets into bounded sets, J'{u) = o(||i^||) as \\u\\ ^ 0 . For p > 0, /^ > 0, let u = u^ ^u~ Mp^^:=

{ueE

^u^ e E,

||^++^-|| 0 for ah Xe[0,l].lfue{ueE: smah enough, we have that

\\u\\ < p}\Mp^^ and i^ ^ 0, then for p > 0

{I'^iu),u+-u-) = (Lu, u+ - U-) + (1 - X){J'{u), u+ - U-) \\J'iu)\\ \\u+ +u-

>ii...+..-inc-(i^Ml« hi

\\u

>0. It fohows that 0 is the unique critical point of /A in {u e E : \\u\\ < p}. By a similar argument, C|(J,0) ^ C|(/o,0) ^ C|(Ji,0) ^ 0, c > 0 such that (Bo) \Hzz{z,t)\ < c(l + \z\') for ah {z,t) G R^^ x R. Suppose that there exist two symmetric 2A^ x 2A^ matrices A(t) and Ao(t) with continuous 27r-periodic entries such that

(10.56)

n{z, t) = ]-A{t)z ' z + G{z, t),

where G'{z^t) = o{\z\) uniformly in t as \z\ -^ oo and

(10.57)

1-L{z, t) = ^Ao{t)z . z + Go{z, t),

where GQ{z,t) = o{\z\) uniformly in t as \z\ -^ 0. We denote by • and I * I the usual inner product and norm in R^^.

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CHAPTER 10. COHOMOLOGY

GROUPS

The Hamiltonian system (HS) satisfying (10.56) and (10.57) is called asymptotically linear both at infinity and at zero. Moreover, it is called nonresonant at infinity if 1 is not a Floquet multiplier of the linear system z = JA(t)z; nonresonance at 0 is defined in a similar way by replacing A{t) with Ao(t). We assume (Bi) \G'{z,t)\ < c ( l + aoo(k|))

for a h z G R ^ ^ and t G R;

(B^) liminf ^.^^^' f; := a^{t) h 0 uniformly for t G R. \z\^oo

Aoo{\z\)

Here and in the sequel, we write a{t) >z 0 if a{t) > 0 and strict inequality holds on a set with positive measure; ttoo, ^oo are the control functions given in the previous section. Let ho : R ^ -^ R ^ be a control function of Go such that

(10.58)

2 < /3o < ^ J 4 T ^ 70 for t smah, ^o(^)

where Ho{t) = J^ ho{s)ds, and /3o, 7 are constants. Obviously, ho{t) = t^ with (5 > 1 is a simple example. Moreover, although ho is defined only for small t > 0, we may assume without loss of generality that it has been extended so that (10.58) holds for ah t G R+. Suppose (B3)

\G'o{^,t)\ < cho{\z\)

for \z\ small;

-\-G^ (z f) z liminf ^^-f-^— := b^{t) h 0 uniformly for t G R. \z\^o Ho[\z\) In order to state our results, we shall need the notation of the > >

2N; 2N; 2N2N.

Theorem 10.40. Assume that H G ^^(R^^ x R, R) satisfies {BQ), {BI), one of the conditions {B^) and A{t) = Ao{t) = 0 (hence 1-C{z,t) = G{z,t) = Go{z,t)). Furthermore, let l-[\z,t) = o{\z\) uniformly in t for \z\ -^ 0. Then {HS) has at least two nontrivial 27r-periodic solutions in each of the following two cases: (1) condition {Bt) holds and either there exists a (5 > 0 such that 1-L{z^t) < 0 whenever \z\ < 6 or (^3), {B^) are satisfied;

256

CHAPTER 10. COHOMOLOGY

GROUPS

(2) condition {B2) holds and either there exists a S > 0 such that 1-L{z^t) > 0 whenever \z\ < S or (^3), (B^) are satisfied.

Let E := ilf 2 (^'^R^^) be the Sobolev space of 27r-periodic R^^- valued functions of the form 00

(10.59)

z(t) = ao +/_^(a/c cos/ct + 6/c sin/ct),

ao, a/c, 6/c G R^^,

k=l 00

such that Yl ^(l^feP + l^feP) < ^^- Then ^ is a Hilbert space with a norm k=l

II • II induced by the inner product (•, •) defined by (10.60)

(z, z') := 27rao • ag + TT ^

k{ak • a'j^ + 6^ • ^fe)-

Let Fk := {a/c cos /ct + bk sin kt : ak^bk G R ^ ^ } ,

A: > 0,

and n

n

^n •= ^ ^ f e = {^ G ^ • ^(0 = «o + y^(afc COS H + 6fc sin kt)}. k=0

k=l

Then ( ^ n ) ^ i is a filtration of E. Denote (10.61)

S' = {En,dn}

with

dn := N{1 ^2n)

=-dimE^.

If 5(t) is a symmetric 2N x 2A^-matrix with continuous 27r-periodic entries, then the operator B given by the formula {Bz,w) := / Jo

B{t)z'wdt

is compact. By Lemma 10.26, the operator LB given by />27r

(10.62)

(L^z, w) := /

is A-proper and M^{LB) (10.63)

{

( - J i - B{t)z) • i(;(it

is well defined and finite. Denote i-{B):=M^{LB), i+{B):=M+{LB) f{B) •=M^{LB)

= =

M^{-LB), dimker(Lij)-

10.6. HAMILTONIAN

SYSTEMS

257

Then we have i-{B)^i^{B)^i^{B)

=0.

Since M^{LB) is in fact the number of linearly independent 27r-periodic solutions of the linear system z = JB(t)z, therefore, 0 < M^(LB) < 2A^. It is well known that under condition (5i), z{t) is a 27r-periodic solution of (HS) if and only if it is a critical point of the C^-functional ^z)

= \ J^"{-Jz

- A{t)z) . zdt - /Q'" G{z,t)dt

(10.64) =

^{Lz,z)-^{z),

which can be rewritten as ^z)

= \ J^"{-Jz

- Ao{t)z) . zdt - /o'" Go{z, t)dt

(10.65) =

^{Loz,z)-^o{z),

where {Lz, z) =

{-Jz - A{t)z) ' zdt, Jo

{Loz,z)=

/

{—Jz —

Ao{t)z)'zdt,

/o PZTT

PZTT

r2-

^f{z)=

i Jo

G{z,t)dt,

'^o{z)=

Go{z,t)dt,

z e E.

Jo

Moreover, ^ G C'^{E,Il) if (BQ) is satisfied. By (10.56)-(10.56), it is easy to check that (10.66)

vl>'(^) = 0(11^11) as | | z | | ^ ^ ,

vl>^(z)=o(||z||)as||2||^0.

In particular, 0 is a trivial solution of (HS). Let L :=

LB,

LQ

:=

LBQ

and introduce a new filtration S := {E'^, (i^}^]^, where E'^ := {R{L) H E^) 0 ker(I/) and d^ = A^(l + 2n) as before. Then L, LQ are A-proper with respect to 8 because they are such with respect to 8^ defined in (10.62). Then (10.67)

M^,iL) =

M^{L)^i-{A)

and (10.68)

M^,{Lo) = M^{Lo) = i-(Ao).

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CHAPTER 10. COHOMOLOGY

GROUPS

We will compute the critical groups C | ( ^ , 0 ) and C|(^,/C(^)). Therefore, we first show how conditions (Bi) and (^2 ) ™ply the {PSy condition with respect to 8. Lemma 10.41. Assume that (^2 ) holds. Then correspondingly, ±

liminf

f^''G(z,t)dt ^Q „ / >Q.

2;Gker(L)

Proof. Note that dimker(I/) < oo, we see that the norm || • || and the L ^ norm are equivalent on ker{L). Moreover, recall that z G ker{L) has the unique continuation property. Therefore, S\\z\\ < \z{t)\ < c||z||, for some S,c> 0 and all t. Recalling the definition of ttoo in (10.49), we have that ^'" ^oo(l^l) ,, 0 ^oo(lkll)

f'-

-

\z\a^i\z\)

Jo INII«oo(||^||)

\\z\\

f

P\\

> f a^{t)dt^ J\z\>M

4|0. Since a^(t) ^ 0 and £ > 0 is arbitrary, the conclusion follows immediately. D

10.6. HAMILTONIAN

SYSTEMS

259

Lemma 10.42. Suppose that (Bi) and (Bf) hold. Then ^ satisfies the (PS)* condition with respect to 8. Moreover, under these hypotheses ^{E^ satisfies the {PS) condition for each n. Proof. Assume that (^2^) holds. Let {zj) be a (P6')*-sequence, i.e., Zj G E'^.^^{zj) is bounded, P'^.^'{zj) -^ 0 and Uj ^ oo as j ^ oo (P^ is the orthogonal projector onto E'^). By Lemma 10.25, we may find a c > 0 and no > 0 such that \\P'^Lz\\ > c\\z\\ for ah z G R{L) n E^ and n > UQ. For z G ^ 4 , write z = w ^ z^ e R{L) n ^n © ker(L). Then (10.69)

p;,^^\zj)

= p;,^Lwj -

p;,^^\zj)

0.

By the definition of ttoo in (10.49), we have that ^ ^ 4 < c,{-r~' for s > aoo{t) H^ t > 0. Therefore, similar to the proof of Lemma 10.41, we get that r27r

aooilz^'lMdt 0 27r

0, we may choose £i > 0 so small that

(10.73)

/

b^{t)dt >l

[

h^{t)dt > 0.

Since kerLo is finite-dimensional, we may assume |^n(OI — c(i?(£i) + c)||z-^|| as long as t G r^. For any £2 > 0, by {H^ ), we have that (10.74)

±^M£!il!l_i!i>5±(i)-e2 J^O[\Zn\)

for all t G r ^ and n large enough. Since Ho{t) is increasing, Ho{\zn\) > Ho{\\zn\\) for |zn| > ll^nll- Since ^ ^

(10.75)

^ 1, we see that

'^"^^^' >

k^WI-K(t)|

,0 1 ^6\\zn-R{s,)\\wr,\

as t G r ^ and n ^ 00, where 5 is as in the proof of Lemma 10.41. Combining (10.72)-(10.75), we have that

for t G r^, \zn{t) < \\zn\\ and n large enough. On the other hand, by (10.72) we get that (10.77)

Ho{\Zn\) ^Odl^nll) r^

< C.

CHAPTER 10. COHOMOLOGY

262

GROUPS

By (10.73), (10.75), (10.76) and (10.77),

r^

(10.78)

> c

b^{t)dt-C£2

^^. > c / =

dt

Ho{\\zn\\j

b^{t)dt-C£2

C-Jo C£2-

Further, we may assume that (^3) holds for ah z. Indeed, suppose that (^3) is satisfied whenever \z\ < SQ. Since ho may be extended such that (10.58) holds for all t, we have by (10.58) that ^(i)/3o-i < M l < ^ ( i ) 7 o - i 7o s ho{s) (3o s

forant>.>0.

It follows that ho{t) > ct^^~^ for t > SQ. Combining this with the asymptotic linearity of the Hamiltonian, we see that (10.79)

\G',{z,t)\ SQ. Keep in mind that (^3) holds for all z. Thus we see that (10.80)

±G'Q{Zn,t)

' Zn\ ^ chQ(\Zn\)\Zn\

Ho{\Zn\)

-

^

Ho{\Zn\

Noting that meas([0, 27r]\r^) < £1, we have that (10.80) implies

±Go(^n,0 Zfi

(10.81)

-dt

[0,27r]\r^

^o(ll^nll) Ho{\Zn\)

< c [0,27r]\r^

dt

Ho{\\Zn\

If U.,1 < WZr. then 5°j!^"[^. < 1. Otherwise, by (10.72),

i^odl^nl

(10.82)

Ho(\Zn\)

, \Zn\ ,^„

10.6. HAMILTONIAN

SYSTEMS

263

Using (10.81)-(10.82) and the Sobolev embedding of E into ^^^^([O, 27r]), we obtain ^Go[Zn,t)

(10.83) /[0,27r]\r^

• Zji^.,

^odl^n

c-ce2-

Jo

eel > 0-

^o(lknll)

Since Si and £2 are arbitrary, we obtain the conclusion of (10.71). Lemma 10.44. Set

{

D

z = z^ ^w e ker(Lo) © (ker(Lo))^, zeE: 0 < \\z\\ 0 and tv ^ (0,1) such that ±{^\z),z^)

for all z G Q(p, K.).

0

E^ := {u e E : u{x, t) =

^

Ujk sin jxe'^^}.

Lemma 10.50. Assume that (C3) and {Cf) hold. Then hminf

G{x,t,v)( ± fo^ G(x, t, v)dxdt ^^ AUM) / ; , / „ : — > 0.

Proof. Note that dim^^ < 00. By the definition of ttoo, we have that

I

Ac.{\v\) dxdt nAoo(lbll) JQ

(5(e)H^;!!}. Since / ^ a^(x, t)dxdt > 0, we may choose an e > 0 so small that (10.94)

/ a^{x,t)dxdt jQ{v,e)

> a^{x,t)dxdt 2 JQ

>0

for any v G E^.

By (Cf), for any £1 > 0, there exists a T(£i) > 0 such that (10.95)

±^(^lM) >a±(x,i)-£i

forany(x,i)efi, |ei>T(£i).

Set 0(1;, £1) Oi(i;,£) ^2(^,£)

{{x,t)en:\v{x,t)\>T{£i)}, {{x,t)en{v,£):\v{x,t)\> {{x,t)en{v,£):\v{x,t)\
0. Therefore, \(J'{u),v)\

that (

[^]

ioTq = M^{Lo)

[

[0]

otherwise;

(

[^]

iovq =

[

[0]

otherwise.

+ MO{Lo),

( C ^ ) impl ies that

In particular,

if bo ^ cr(5), conclusions

M^{Lo),

(1) and (2) still hold with M^ {Lo) = 0.

P r o o f . (1) We first consider t h e case of bo G CF{B). W i t h o u t loss of generality, we may assume in {CQ) t h a t a < 0 < 6. Write u = u^ ^ u~ ^ u^ with u^ e E^.vP e E^. Recalling (10.91), we observe t h a t (10.99)

{I'{u),u-^

-u-

= {Lou'^jU'^)

-u^) — {Lou~,u~)-\-

/ go{x^t^u){u^

— u~ — u^)dxdt

JQ

> / {K.'^{u'^f

^ K.~{u~f

^ go{x,t,u){u'^

- u~ -

u^))dxdt.

JQ

Now we estimate t h e integrand in (10.99). Define (10.100)

Qi

{{x,t)

e n :u{x,t)

= 0};

(10.101)

O2

{{x,t)

G n :u{x,t)

j^O,\u-

(10.102)

O3

{{x,t)

G O : 0 < \u{x,t)\

(10.103)

O4

{{x,t)

en-.ao

-^u^\ < |i^+|};

> |^+|}; \u^\}.

If {x,t) G Oi in (10.100), then (10.104)

K.-^{u-^f^K.-{u-f^go{x,t,u){u-^-u-

- u^) > 0.

If ( x , t ) G O2 in (10.101), then (10.105)

-u{u-^

-u-

-u^)

a{u-^f -a{u-

- u^) ^u^f

Hence, (10.107)

/^+(^+)2 + K.-{u-f

+ go{x, t, u){u-^ - u' - u^)

>0. If {x,t) G O3 in (10.102), then -u{u~^ -u~

- u^) > 0.

By (Cf), (10.108)

gQ{x,t,u){u'^ -u-

-u^)>{)

>d{u'^f.

Also we have that K.'^{u'^f ^ K.- {u-f

(10.109)

^ gQ{x,t,u){u^

- u' - u^) > 0.

If (x,t) G O4 in (10.103), by (CQ) we have (10.110)

go{x, t, u){u'^ -u-

- u^)

>-6(^-+^0)^ consequently, (10.111)

/^+(^+)^ + K-{u-f + go{x,t,u){u'^ -u> /^+(^+)2 + K-{u-f - b{u- + u^f > K^{u^f

+ {K- - b){u-f

- b{uy

- u^)

- 26|^%-|.

Note that dim^Q < ^^- Hence there exists a /3o > 0 such that sup{|^0(x,t)| : {x,t) G 0 } < /3o||^^||, Choose

1

. rmin{/^+,/^

for ah u^ e E^.

- 6}crg(/^ - 6 )

CTQ 1

282

CHAPTER 10. COHOMOLOGY

GROUPS

Then for ||i^|| < po, we have that |i^^|


0.

Combining (10.99)-(10.113), {I'iu),u+ -u-

-u^)>0

for||u|| 0 such that 0 < s < ^So{so comes from (^2)) and that maxj/c^ - f

:k^ -f

0,k e ZJ G N } + e = min{A:^ - j ^ ^ s : k'^ - j ^ ^ s > 0,k e ZJ G N } . Let / ( x , t, ^) = ^o(^, ^, 0 = ^^ + ^o(^, ^, 0 - ^^ •= ^^ + ^o(^, t, C). By Lemma 10.57,

where L^ is defined as {LsU, u) = I {u^ — u^^ + su )dxdt. Ji Then the negative space of L^ is E~ := {u e E : u{x, t) =

^

Ujk sin jxe'^^}.

Hence dim(^~ H^^) = ^ when £ is small enough. Consequently, Mg{Ls) = 0 and C | ( / , 0 ) ^ (^g,o[^], yqeZ. D

Proof of Theorem 10.47. In fact, if X:(/) = {0}, then

c|(/,o) = c|(/,x:(/)). Then Theorem 10.47 follows immediately from Lemma 10.55 and Lemma 10.56. D

284

CHAPTER 10. COHOMOLOGY

GROUPS

Proof of Theorem 10.48. From Lemma 10.56, Lemma 10.57 and Lemma 10.58 we get the existence of a nontrivial solution ui ^ 0 from the fact that C|(/,0)^C|(J,/C)

for some q. Moreover, suppose that ui is nondegenerate, i.e..

In order to deal with all cases simultaneously, we assume that

If there is no other critical point, then by the Morse inequalities, we have

(-ir[i] + (-ir[i] = (-ir~[i], a contradiction.

D

Proof of Theorem 10.49. For any £ > 0, there exists co^ > 0 such that ±{u;g{x,t,u;) -2G{x,t,uo))

< (a^(x,t) + £)|cj|^+^

for \uo\ >uo^.

Therefore, da /±G{x,t,u;)\ / duo \


ujs'

Integrating the above inequality over the interval [c(;,C(;i] C [cj^, oo) yields the estimate

fG{x,t,ui) ±

G{x,t,u)\

2

9

(a^{x,t)^£) —

^

^_^ V^l

^_^ ~^

'

Therefore, cj^oo

CJ "^^

1 — cr

Similarly, the above limits are also true if cj -^ — oo. We have shown that [Cf) implies [Cf) with a^(t) = |t|^ and a G (0,1). D ^4

Notes and Comments. When / is superlinear, (B) was studied in F. C. Chang-L. Sanchez [92] and G. Feireisl [151]. Several papers have dealt with (B) in case that / is asymptotically linear, see for examples: T. Bartsch-Y.

10.7. ASYMPTOTICALLY

LINEAR BEAM EQUATIONS

285

Ding [36] and D. Lupo-A. M. Micheletti [246, 247]. These papers were done under the assumption that g is bounded globahy and satisfies the condition of Ahmad-Lazer-Paul type (i.e., G{x,t,^) -^ oo (or —oo) as \^\ -^ oo uniformly in (x, t)). Their main tool was a minimax argument (linking and limit relative category). Lemma 10.51 was originally proved in W. Kryszewski-A. Szulkin [200] for wave equations (see also K. Tanaka [365]). We refer the readers to A. C. Lazer-P. J. Mckenna [206, 207], J. Liu [231, 232], A. M. Micheletti-C. Saccon [257] and A. M. Micheletti-A. A. Pistoia-C. Saccon [256] for beam equations via linking type arguments; H. Brezis-L. Nirenberg [68], H. Brezis-J. M. Coron-L. Nirenberg [69], P. Rabinowitz [291], W. KryszewskiA. Szulkin[200] and S. Li-A. Szulkin [215] and the references cited therein for wave equations. The main results of this section were established in W. Zou [389]. Assumptions (Cg ) are a generalization of the so-called nonquadratic conditions considered by D. G. Costa, C. A. Magalhaes, E. A. B. Silva, etc. (see [107, 111, 112, 113, 339]), which were used in T, Bartsch-M. Clapp [32] to deal with superlinear noncooperative elliptic systems. By linking arguments, some results were established for the study of strongly indefinite functionals. One of the most important parts of the theory was developed by V. Benci-P. H. Rabinowitz [55]. In E. A. B. Silva [339], the framework introduced in [55] was used to prove the existence of subharmonic periodic solutions for a class of asymptotically quadratic first order Hamiltonian systems satisfying the generalized version of the LandesmanLazer condition introduced in E. A. B. Silva [336] (see also D. G. Costa[107] and D. G. de Figueiredo-L Massab6[161]). In [336, 340], E. A. B. Silva has also established some abstract critical point theorems to study the existence and the multiplicity of critical points for strongly indefinite functionals of the form ^(Lu^u) -\- J{u), with J{u) unbounded and satisfying the (PS)* condition. An earlier result on strongly indefinite functionals with applications can also be found in H. Hofer [187]. The result of [340] is used to establish the existence of nonzero solutions for noncooperative elliptic systems (cf. E. A. B. Silva [341]). We also refer readers to N. Ghoussoub's duality and perturbation methods in critical point theory (see [170, 171]) which involve some minmax principles with relaxed boundary conditions and to A. C. Lazer-S. Solimini [208], M. Ramos-L. Sanchez [299], K. Perera-M. Schechter [287] and S. Solimini [345] for Morse index estimates in minimax theorems.

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Index {PS)*, 217, 218

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