Morse Theoretic Aspects of p-Laplacian Type Operators (Mathematical Surveys and Monographs)

Morse Theoretic Aspects of p-Laplacian Type Operators Kanishka Perera Ravi P. Agarwal

Donal O'Regan

American Mathematical Society

Morse Theoretic Aspects of p-Laplacian Type Operators

Mathematical Surveys and

Monographs

Volume 161

Morse Theoretic Aspects of p-Laplacian Type Operators Kanishka Perera Ravi P. Agarwal

Donal O'Regan

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen, Chair J. T. Stafford Benjamin Sudakov 2000 Mathematics Subject Classification. Primary 58E05, 47305, 47310, 35360.

For additional information and updates on this book, visit

www.arns.org/bookpages/surv-161

Library of Congress Cataloging-in-Publication Data Perera, Kanishka, 1969Morse theoretic aspects of p-Laplacian type operators / Kanishka Perera, Ravi Agarwal, Donal O'Regan. p. cm. (Mathematical surveys and monographs v. 161) Includes bibliographical references ISBN 978-0-8218-4968-2 (alk. paper)

1. Morse theory. 2. Laplacian operator. 3. Critical point theory (Mathematical analysis) I. Agarwal, Ravi, 1947- It. O'Regan, Donal. III. Title QA614.7.P47

2010 2009050663

515'.3533-dc22

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Contents vii

Preface

An Overview

ix

Chapter 0. Morse Theory and Variational Problems 0.1. Compactness Conditions 0.2. Deformation Lemmas 0.3. Critical Groups 0.4. Minimax Principle 0.5. Linking 0.6. Local Linking 0.7. p-Laplacian Chapter 1. Abstract Formulation and Examples 1.1. p-Laplacian Problems 1.2. AP Laplacian Problems 1.3. Problems in Weighted Sobolev Spaces 1.4. q-Kirchhoff Problems 1.5. Dynamic Equations on Time Scales 1.6. Other Boundary Conditions 1.7. p-Biharmonic Problems 1.8. Systems of Equations Chapter 2. Background Material 2.1. Homotopy 2.2. Direct Limits 2.3. Alexander-Spanier Cohomology Theory 2.4. 2.5.

Principal 712-Bundles Cohomological Index

Chapter 3. Critical Point Theory 3.1. Compactness Conditions 3.2. Deformation Lemmas 3.3. Minimax Principle 3.4. Critical Groups 3.5. Minimizers and Maximizers 3.6. Homotopical Linking 3.7.

Cohomological Linking V

CONTENTS

vi

3.8. Nontrivial Critical Points 3.9. Mountain Pass Points 3.10. Three Critical Points Theorem 3.11. Cohomological Local Splitting 3.12. Even Functionals and Multiplicity 3.13. Pseudo-Index 3.14. Functionals on Finsler Manifolds Chapter 4.

4.1. 4.2. 4.3.

p-Linear Eigenvalue Problems

Variational Setting Minimax Eigenvalues Nontrivial Critical Groups

Chapter 5. Existence Theory 5.1. p-Sublinear Case 5.2. Asymptotically p-Linear Case 5.3. p-Superlinear Case Chapter 6.

Monotonicity and Uniqueness

Chapter 7. Nontrivial Solutions and Multiplicity 7.1. Mountain Pass Solutions 7.2. Solutions via a Cohomological Local Splitting 7.3. Nonlinearities that Cross an Eigenvalue 7.4. Odd Nonlinearities

89 89 90 91 93

Chapter 8. Jumping Nonlinearities and the Dancer-Fucik Spectrum 8.1. Variational Setting 8.2. A Family of Curves in the Spectrum 8.3. Homotopy Invariance of Critical Groups 8.4. Perturbations and Solvability

97 99 100 102 105

Chapter 9. Indefinite Eigenvalue Problems 9.1. Positive Eigenvalues 9.2. Negative Eigenvalues 9.3. General Case 9.4. Critical Groups of Perturbed Problems

109 109 111 112 113

Chapter 10. Anisotropic Systems 10.1. Eigenvalue Problems 10.2. Critical Groups of Perturbed Systems 10.3. Classification of Systems

117 119 125 128

Bibliography

135

Preface The p-Laplacian operator

Apu = div

(IVulr-2 Vu),

PC (1, GO)

arises in non-Newtonian fluid flows, turbulent filtration in porous media, plasticity theory, rheology, glaciology, and in many other application areas; see, e.g., Esteban and Vazquez [48] and Padial, Takae, and Tello [90]. Problems involving the p-Laplacian have been studied extensively in the literature during the last fifty years. However, only a few papers have used Morse theoretic methods to study such problems; see, e.g., Vannella [130], Cingolani

and Vannella [29, 31], Dancer and Perera [40], Liu and Su [74], Jiu and Su [58], Perera [98, 99, 100], Bartsch and Liu [15], Jiang [57], Liu and Li [75], Ayoujil and El Amrouss [10, 11, 12), Cingolani and Degiovanni [30], Guo and Liu [55], Liu and Liu [73], Degiovanni and Lancelotti [43, 44], Liu and Geng [70], Tanaka [129], and Fang and Liu [50]. The purpose of this

monograph is to fill this gap in the literature by presenting a Morse theoretic study of a very general class of homogeneous operators that includes the p-Laplacian as a special case. Infinite dimensional Morse theory has been used extensively in the literature to study semilinear problems (see, e.g., Chang [28] or Mawhin and Willem [81]). In this theory the behavior of a Cl-functional defined on a Banach space near one of its isolated critical points is described by its critical groups, and there are standard tools for computing these groups for the variational functional associated with a semilinear problem. They include the Morse and splitting lemmas, the shifting theorem, and various linking and local linking theorems based on eigenspaces that give critical points with nontrivial critical groups. Unfortunately, none of them apply to quasilinear problems where the Euler functional is no longer defined on a Hilbert space or is C2 and there are no eigenspaces to work with. We will systematically develop alternative tools, such as nonlinear linking and local linking theories, in order to effectively apply Morse theory to such problems. A complete description of the spectrum of a quasilinear operator such as the p-Laplacian is in general not available. Unbounded sequences of eigenvalues can be constructed using various minimax schemes, but it is generally not known whether they give a full list, and it is often unclear whether different schemes give the same eigenvalues. The standard eigenvalue sequence based on the Krasnoselskii genus is not useful for obtaining nontrivial critical vii

viii

PREFACE

groups or for constructing linking sets or local linkings. We will work with a new sequence of eigenvalues introduced by the first author in [98] that uses the 7Z2-cohomological index of Fadell and Rabinowitz. The necessary background material on algebraic topology and the cohomological index will be given in order to make the text as self-contained as possible. One of the main points that we would like to make here is that, contrary to the prevailing sentiment in the literature, the lack of a complete list of eigenvalues is not a serious obstacle to effectively applying critical point theory. Indeed, our sequence of eigenvalues is sufficient to adapt many of the standard variational methods for solving semilinear problems to the quasilinear case. In particular, we will obtain nontrivial critical groups and use the stability and piercing properties of the cohomological index to construct

new linking sets that are readily applicable to quasilinear problems. Of course, such constructions cannot be based on linear subspaces since we no longer have eigenspaces. We will instead use nonlinear splittings based on certain sub- and superlevel sets whose cohomological indices can be precisely calculated. We will also introduce a new notion of local linking based on these splittings. We will describe the general setting and give some examples in Chap-

ter 1, but first we give an overview of the theory developed here and a preliminary survey chapter on Morse theoretic methods used in variational problems in order to set up the history and context.

An Overview Let D be a C1-functional defined on a real Banach space W and satisfying the (PS) condition. In Morse theory the local behavior of 4' near an isolated critical point u is described by the sequence of critical groups (1)

Cq ($, u) = Hq(,D` n U, $` n U\ {u}),

q >, 0

where c = 'F(u) is the corresponding critical value, 4'` is the sublevel set {u e W : f(u) 5 c}, U is a neighborhood of u containing no other critical points, and H denotes cohomology. They are independent of U by the excision property. When the critical values are bounded from below, the global behavior of 4' can be described by the critical groups at infinity Cq(1D, co) = Hq(W,1"),

q?0

where a is less than all critical values. They are independent of a by the second deformation lemma and the homotopy invariance of cohomology groups. When 4' has only a finite number of critical points u1, ... , Uk, their critical groups are related to those at infinity by k

rank Cq(4',uj) >, rank Cq(4',co)

Vq

a=1

(see Proposition 3.16). Thus, if C"(4', oo) # 0, then 4' has a critical point u with Cq(4',u) # 0. If zero is the only critical point of (F and 4) (0) = 0, then taking U = W in (1), and noting that (F° is a deformation retract of W and 4'°\ {0} deformation retracts to (DI by the second deformation lemma, gives

Cq(4',0)=Hq(4'°,4'°\{0})

Hq(W,,D°\{0})

z Hq(W,,Da) = Cq(4', ao)

Vq.

Thus, if Cq(4', 0) 4 Cq((F, co) for some q, then D has a critical point u # 0. Such ideas have been used extensively in the literature to obtain multiple nontrivial solutions of semilinear elliptic boundary value problems (see, e.g., Mawhin and Willem [81], Chang [28], Bartsch and Li [14], and their references). Now consider the eigenvalue problem

OP u = Alujp-2 u in S2

i

u=0

onOQ ix

AN OVERVIEW

x

where 11 is a bounded domain in ir, n > 1, Op u = div (I

Vulp-2 Du)

is the p-Laplacian of it, and p e (1, oo). The eigenfunctions coincide with the critical points of the C1-functional 'Pa(u) =

f (pulp - A luIP sz

defined on the Sobolev space Wo'p(El) with the usual norm

\i

p r hull = Cfst lvulp)

.

When A is not an eigenvalue, zero is the /only critical point of 4?A and we may take U = {u e Wo'p(S2) : DVII i} in the definition (1). Since 4Pa is positive homogeneous, 4ia n U radially contracts to the origin and 4?0 n U\ {0} radially deformation retracts onto

4°nS=qja where S is the unit sphere in WW'p(Kl) and 1P (U) =

1

u e S.

,

fn lul' It follows that bqO 9,

where 6 is the Kronecker delta, 9 is the coefficient group, and H denotes reduced cohomology. Note also that the eigenvalues coincide with the critical values of 'Ir by the Lagrange multiplier rule.

In the semilinear case p = 2, the spectrum a(-0) consists of isolated eigenvalues Ak, repeated according to their multiplicities, satisfying

O A2. Thus, the question arises as to whether there is a nontrivial critical group when A > A2. An affirmative answer was given in Perera [98] where a new sequence of eigenvalues was constructed using a minimax scheme involving the Z2-cohomological index of Fadell and Rabinowitz [49] as follows.

Let .F denote the class of symmetric subsets of S, let i(M) denote the cohomological index of M e F, and set Ak :=

inf

MGY

i(M)3k

sup 111(u).

"M

Then Ak / co is a sequence of eigenvalues, and if Ak < Ak+l, then (5)

i(WAk) = i(S\tak+') = k

where

= {u c S :'I (u) C Ak}, WAk+' = {u E S :'1(u) i Ak+1} (see Theorem 4.6). Thus, if Ak < A < Ak+1, then peak

i(41A) = k

by the monotonicity of the index, which implies that Hk-1(41a) # 0

AN OVERVIEW

xii

(see Proposition 2.14) and hence Ck((Da, 0) # 0

(6)

by (2). The structure provided by this new sequence of eigenvalues is sufficient

to adapt many of the standard variational methods for solving semilinear problems to the quasilinear case. In particular, we will construct new linking sets and local linkings that are readily applicable to quasilinear problems. Of course, such constructions cannot be based on linear subspaces since we no longer have eigenspaces to work with. They will instead use nonlinear splittings generated by the sub- and superlevel sets of ' that appear in (5), and the indices given there will play a key role in these new topological constructions as we will see next. Consider the boundary value problem DP v. = f (x, u) (7)

u=0

in Q

onOQ

where the nonlinearity f is a Caratheodory function on 11 x lR satisfying the subcritical growth condition I f(x, t)I - 0. Then max $(A) < inf 4) (B)

(8)

if R is sufficiently large, and A cohomologically links B in dimension k - 1 in the sense that the homomorphism Hk-1(A)

Hk-1(H1(Q)\B) -' induced by the inclusion is nontrivial. So it follows that problem (7) has a solution it with u) # 0 (see Proposition 3.25). We may ask whether this well-known argument can be modified to obtain

the same result in the quasilinear case p # 2 where we no longer have the splitting given in (4). We will give an affirmative answer as follows. Let

A={Ru:aeW },

B={tu:ue 111 4+1+t>0}

with R > 0. Then (8) still holds if R is sufficiently large, and A cohomologically links B in dimension k - 1 by (5) and the following theorem proved in Section 3.7, so problem (7) again has a solution it with Ck(1,'a) # 0.

THEOREM 1. Let A0 and B0 be disjoint nonempty closed symmetric subsets of the unit sphere S in a Banach space such that

i(Ao) = i(S\Bo) = k where i denotes the cohomological index, and let

A={Ru:acAo},

B={tu:ueBo,t>0}

with R > 0. Then A cohomologically links B in dimension k - 1.

Now suppose f (x, 0) = 0, so that problem (7) has the trivial solution u(x) = 0. Assume that (9)

lim f (x, t) = A, uniformly in x E Q,

(10)

pF(x,t) > Ak+I ItID V(x,t) E Q x 1

t,O ItIp-2 t Ak < A < Ak+l, and the sign condition

AN OVERVIEW

x'v

holds. In the p-superlinear case it is customary to also assume the following Ambrosetti-Rabinowitz type condition to ensure that 'F satisfies the (PS) condition:

0 < p F(x, t) S t f (x, t) Vx e 52, Itl large

(11)

for some p > p.

In the semilinear case p = 2, we can then obtain a nontrivial solution of problem (7) using the well-known saddle point theorem of Rabinowitz as follows. Fix a wo e H+\{0} and let

X

{u = v + s w0 : v e H-, s; 0, 1IuIl S R},

A={veH-:IIvII

R} u{ueX: hull =R},

B = {weH+:IIwII =r} with H± as in (4) and R > r > 0. Then max4 (A) < 0 < inf 4)(B)

(12)

if R is sufficiently large and r is sufficiently small, and A homotopically links

B with respect to X in the sense' that

-y(X)nB#0 V7er where

r={ryeC(X,Ho(Q)):'YIA=idA}. So it follows that

e := inf sup

yeV ue7(X)

4>(v)

is a positive critical level of 115 (see Proposition 3.21). Again we may ask whether linking sets that would enable us to use this

argument in the quasilinear case p # 2 can be constructed. In Perera and Szulkin [105] the following such construction based on the piercing property

of the index (see Proposition 2.12) was given. Recall that the cone CA0 on a topological space A0 is the quotient space of A0 x [0, 1] obtained by collapsing A0 x {1} to a point. We identify A0 x {0} with A0 itself. Fix an h e C(CTAk, S) such that h(Cq/Pk) is closed and hl,wAk = idq,Ak, and let

X = {tu:neh(CTAk), 0k

We will again show that ak J' +oo and ak -oo are sequences of positive and negative eigenvalues, respectively, and if ak < ak+1 (resp. ak+1 < ak ), then k

(resp. 2((`p )ak) = i(M \(il/ )ak+i) = k), in particular, if ak < A < ak+1 or Ak+1 < A < ak, then Ck(,Da, 0) # 0.

This will allow us to extend our existence and multiplicity theory for a single equation to systems. For example, suppose F(x, 0) -- 0, so that the system (15) has the trivial

solution u(x) = 0. Assume that F(x, u) = A J(x, u) + G(x, u) where A is not an eigenvalue of (16) and In

IG(x,u)I s C

Iuils,

V(x,u) e Q

i=1

for some si e (pi, p,*). Further assume the following superlinearity condition:

there are

lei

> Pi, i = 1, ... , m such that

r1

Z` i=1 Pi

1 \ aF -) ui - is bounded aui jui

from below and Ui OF

0 < F(x, u) i=1

lii aui

(x, u)

Vx e li,

IuI

large.

We will obtain a nontrivial solution of (15) under these assumptions in Sections 10.2 and 10.3.

AN OVERVIEW

All this machinery can be adapted to many other p-Laplacian like operators as well. Therefore we will develop our theory in an abstract operator setting that includes many of them as special cases.

CHAPTER 0

Morse Theory and Variational Problems In this preliminary chapter we give a brief survey of Morse theoretic methods used in variational problems. General references are Milnor [84], Mawhin and Willem [81], Chang [28], and Benci [19]. We consider a real-valued function 4' defined on a real Banach space (W, 11[I). We say that 4' is FYechet differentiable at it e W if there is an element 4'(u) of the dual space (W*, II.[1*), called the FYechet derivative of 4) at it, such that

4'(u+v)=4'(u)+(4"(u), v)+o(IIvIj) as v->0 in W, where is the duality pairing. The functional 4' is continuously Frechet differentiable on W, or belongs to the class Cr (W, R), if 4' is defined every-

where and the map W -> W*, it 4'(u) is continuous. We assume that 4' e Cr (W, 1R) for the rest of the chapter. We say that it is a critical point of 4' if 4"(u) = 0. A real number c e F(W) is a critical value of 4' if there is a critical point it with 4'(u) = c, otherwise it is a regular value. We use the notations 4% = {u W D(u) >, a}, 4'b

=

{n W : 4'(u) < b},

4)a = 4'

4,b,

K={ueW:"(u)=0}, W=W\K, Kb=Kn0", K`=Kc for the various superlevel, sublevel, critical, and regular sets of 4'. We begin by recalling the compactness condition of Palais and Smale and its weaker variant given by Cerami in Section 0.1. Then we state the first and second deformation lemmas under the Cerami's condition in Section 0.2. In Section 0.3 we define the critical groups of an isolated critical point and summarize the basic results of Morse theory. These include the Morse inequalities, Morse lemma and its generalization splitting lemma, shifting theorem of Gromoll and Meyer, and the handle body theorem. Next we discuss the minimax principle in Section 0.4. Section 0.5 contains a discussion of homotopical linking, pairs of critical points with nontrivial critical groups produced by homological linking, and nonstandard geometries without a finite dimensional closed loop. We recall the notion of local linking and an

alternative for a critical point produced by a local linking in Section 0.6. We conclude with a result on nontrivial critical groups associated with the p-Laplacian in Section 0.7.

i

0. MORSE THEORY AND VARIATIONAL PROBLEMS

2

0.1. Compactness Conditions It is usually necessary to assume some sort of a "compactness condition" when seeking critical points of a functional. The following condition was originally introduced by Palais and Smale [91]: 4) satisfies the Palais-Smale compactness condition at the level c, or (PS)0 for short, if every sequence

(uj) c W such that c, 4)'(uj) - 0, lp(ui) called a (PS)0 sequence, has a convergent subsequence; 4) satisfies (PS) if it satisfies (PS)0 for every c e 1W, or equivalently, if every sequence such that 4)(u) is bounded and 4)'(u) 0, called a (PS) sequence, has a convergent subsequence. The following weaker version was introduced by Cerami [25]: 4) satisfies the Cerami condition at the level c, or (C)0 for short, if every sequence such

that

iD(ui) -c,

(1 + IIu1II) 4''(ui) -0, called a (C)0 sequence, has a convergent subsequence;'P satisfies (C) if it satisfies (C), for every c, or equivalently, if every sequence such that P(ub) is bounded and (1 + IIuj 11) 4)'(ui) --. 0, called a (C) sequence, has a convergent subsequence. This condition is weaker since a (C)0 (resp. (C)) sequence is clearly a (PS)0 (resp. (PS)) sequence also. The limit of a (PS)0 (resp. (PS)) sequence is in K° (resp. K) since 4) and 42' are continuous. Since any sequence in K` is a (C)0 sequence, it follows that K° is a compact set when (C)0 holds.

0.2. Deformation Lemmas An essential tool for locating critical points is the deformation lemmas, which allow to lower sublevel sets of a functional, away from its critical set. The main ingredient in their proofs is a suitable negative pseudo-gradient flow, a notion due to Palais [93]: a pseudo-gradient vector field for 4) on W is a locally Lipschitz continuous mapping V : W - W satisfying

IV(u)II 5 j41'(u)jj*, 2(4)'(u),V(u)) > (1(,(u)D*)2 Vue W. Such a vector field exists, and may be chosen to be odd when 4) is even. The first deformation lemma provides a local deformation near a (possibly critical) level set of a functional.

Lemma 0.1 (First Deformation Lemma). If c e W, C is a bounded set containing K`, 5, k > 0, and 4) satisfies (C)0, then there are an Co > 0 and, for each e e (0,eo), a map rl e C([0, 1] x W,W) satisfying

(i) n(0,) = idw,

(ii) rt(t,) is a homeomorphism of W for all t e [0,1], (iii) rl(t, ) is the identity outside A = 4,12E\Na 3(C) for all t e [0, 1], (iv) I ki(t, u) - A < (1 + jjujj ) 5/k V(t, u) E [0, 1] X W, (v) 4)(?7(., u)) is nonincreasing for all u c W,

0.3. CRITICAL GROUPS

3

(vi) q7(1 4)c+e\Nb(C)) e $C-F.

When 4) is even and C is symmetric, 77 may be chosen so that q(t, ) is odd for all t e [0,1]. First deformation lemma under the (PS), condition is due to Palais [92]; see also Rabinowitz [108]. The proof under the (C), condition was given by Cerami [25] and Bartolo, Benci, and Fortunato [13]. The particular version given here will be proved in Section 3.2. The second deformation lemma implies that the homotopy type of sublevel sets can change only when crossing a critical level.

Lemma 0.2 (Second Deformation Lemma). If -oo < a < b < +eo and 4) has only a finite number of critical points at the level a, has no critical values

in (a, b), and satisfies (C), for all c e [a, b] n R, then 4p° is a deformation retract of 4)6\K6, i.e., there is a map q e C([0,1] x (4)b\Kb), 4)b\Kb), called a deformation retraction of 4)6\K6 onto 4)a, satisfying (i) q(0, ) = 2dpb\Kb, (ii) q(t, idta Vt e [0, 1],

(iii) q(1, ,b\Kb) =

p°

Second deformation lemma under the (PS), condition is due to Rothe [117], Chang [27], and Wang [131]. The proof under the (C), condition can be found in Bartsch and Li [14], Perera and Schechter [104], and in Section 3.2.

0.3. Critical Groups In Morse theory the local behavior of lb near an isolated critical point u is described by the sequence of critical groups (0.1)

Cq((D, u) = Hq(4)` n U, V n U\ {u}),

q >, 0

where c = 4)(u) is the corresponding critical value, U is a neighborhood of u containing no other critical points, and H denotes singular homology. They are independent of the choice of U by the excision property. For example, if it is a local minimizer, C. ((D, u) = 5qo B where S is the

Kronecker delta and 9 is the coefficient group. A critical point it with Cl (4), u) # 0 is called a mountain pass point.

Let -co < a < b C +co be regular values and assume that 4) has only isolated critical values cl < c2 < in (a, b), with a finite number of critical points at each level, and satisfies (PS), for all c e [a, b] n R. Then the Morse type numbers of 4) with respect to the interval (a, b) are defined by Mq(a, b) _

rank

q)0

0. MORSE THEORY AND VARIATIONAL PROBLEMS

where a = a1 < cl < a2 < c2 < They are independent of the ai by the second deformation lemma, and are related to the critical groups by Mq(a,b) _ Y, rank C,, ((b, u). Writing fj (a, b) = rank Hj (4'6, 4'a), we have

Theorem 0.3 (Morse Inequalities). If there is only a finite number of critical points in 4b' then q

E

q

(-1)q-iMj ? ) (-1)q J$

j=0

and

q > 0,

j=o

X

E (-1)jMj j=0 when the series converge.

OD

=

j=0

(-10j

Critical groups are invariant under homotopies that preserve the isolatedness of the critical point; see Rothe [116], Chang and Ghoussoub [26], and Corvellec and Hantoute [33].

Theorem 0.4. If fit, t e [0,1] is a family of C'-functionals on W satisfying (PS), u is a critical point of each 4't, and there is a closed neighborhood U such that (i) U contains no other critical points of 4't, (ii) the map [0, 1] C1(U, k), t - 'Dt is continuous, then C. (4t, u) are independent of t. When the critical values are bounded from below and 4) satisfies (C), the global behavior of P can be described by the critical groups at infinity introduced by Bartsch and Li [14]

Cq(', co) = Hq(W, r), q% 0 where a is less than all critical values. They are independent of a by the second deformation lemma and the homotopy invariance of the homology groups.

For example, if 4' is bounded from below, C. (4', co) = Sqo G. If 4' is unbounded from below, Cq(4', co) = Hq_1(,a) where H denotes the reduced groups.

Theorem 0.5. If Ck(4, co) # 0 and 4' has only a finite number of critical points and satisfies (C), then D has a critical point u with Ck(4', u) # 0. The second deformation lemma implies that Cq((D, w) = Cq(4', 0) if u = 0 is the only critical point of 4', so $ has a nontrivial critical point if Cq(4', 0) # Cq(4, cc) for some q.

0.3. CRITICAL GROUPS

5

Now suppose that W is a Hilbert space (H, and 4) e C2(H, IR). Then the Hessian A = 4)"(u) is a self-adjoint operator on H for each u. When u is a critical point the dimension of the negative space of A is called the Morse index of u and is denoted by m(u), and m* (u) =m(u) +dim ker A

is called the large Morse index. We say that u is nondegenerate if A is invertible. The Morse lemma describes the local behavior of the functional near a nondegenerate critical point.

Lemma 0.6 (Morse Lemma). If u is a nondegenerate critical point of $, then there is a local diffeomorphism 1; from a neighborhood U of u into H with Z;(u) = 0 such that C( -1(v)) = 4)(u) + Z (Av, v)

,

v e (U).

Morse lemma in R' was proved by Morse [85]. Palais [92], Schwartz [123], and Nirenberg [88] extended it to Hilbert spaces when 4) is C3. Proof in the C2 case is due to Kuiper [62] and Cambini [23]. A direct consequence of the Morse lemma is

Theorem 0.7. If u is a nondegenerate critical point of 4', then Ce(4); u)

5gm(v.) G-

The handle body theorem describes the change in topology as the level sets pass through a critical level on which there are only nondegenerate critical points.

Theorem 0.8 (Handle Body Theorem). If c is an isolated critical value of ID for which there are only a finite number of nondegenerate critical points ui, i = 1, ... , k, with Morse indices mi = m,(uj), and 4) satisfies (PS), then there are an e > 0 and homeomorphisms oj from the unit disk D'i in 1[8m= into H such that n ya(Dm`) _ -1(c - e) n = Wti(OD'i) 4)C-6

yj(Dm`)

and D`-E U Ua=I wj(Dmi) is a deformation retract of D`+E

The references for Theorems 0.3, 0.7, and 0.8 are Morse [85], Pitcher [106], Milnor [84], Rothe [114, 115, 117], Palais [92], Palais and Smale [91], Smale [124], Marino and Prodi [79], Schwartz [123], Mawhin and Willem [81], and Chang [28]. The splitting lemma generalizes the Morse lemma to degenerate critical points. Assume that the origin is an isolated degenerate critical point of $ and 0 is an isolated point of the spectrum of A = 4)"(0). Let N = ker A and

write H = NQ N1,u=v+w. Lemma 0.9 (Splitting Lemma). There are a ball B c H centered at the origin, a local homeomorphism e from B into H with 1f (0) = 0, and a map

rl e C'(B n N, N1) such that (Aw, w) + D(v + i7 (v)),

'D

2

u e B.

0. MORSE THEORY AND VARIATIONAL PROBLEMS

6

Splitting lemma when A is a compact perturbation of the identity was proved by Gromoll and Meyer [54] for p e C3 and by Hofer [56] in the C2

case. Mawhin and Willem [80, 81] extended it to the case where A is a Fredholm operator of index zero. The general version given here is due to Chang [28]. A consequence of the splitting lemma is

Theorem 0.10 (Shifting Theorem). We have Cq (`h, 0) = Cq-m(0) (4IN , O)

Vg

where M = . (B n N) is the degenerate submanifold of P at 0. Shifting theorem is due to Gromoll and Meyer [54]; see also Mawhin and Willem [81] and Chang [28]. Since dim.IV = m*(0) - m(0), shifting theorem gives us the following Morse index estimates when there is a nontrivial critical group.

Corollary 0.11. If Ck(4),0) # 0, then m(0) 0 there is a C2 functional 4E on H such that (2) 110- - 4)IIC2(H) S E,

(ii) 4)e = in H\ Ua 1 BE(uj), (iii) (b, has only nondegenerate critical points in BE(uj) and their Morse indices are in [m(ui),m*(ui)], (iv) 4) satisfies (PS) 4E satisfies (PS).

0.4. MINIMAX PRINCIPLE

7

0.4. Minimax Principle Minimax principle originated in the work of Ljusternik and Schnirelmann [76] and is a useful tool for finding critical points of a functional. Note that the first deformation lemma implies that if c is a regular value and 'F satisfies (C)°, then the family D°, E of maps ?7 e C([0,1] x W, W) satisfying (i) 77(0,) = idw, (ii) 77(t, ) is a homeomorphism of W for all t c [0, 1], (iii) 7)(t, -) is the identity outside Dc±2e for all t c [0,1], (iv) ID (?7(, u)) is nonincreasing for all u c W, (v)

7l(1,'C-V-E) C'FC-E

is nonempty for all sufficiently small a > 0. We say that a family T of subsets of W is invariant under D°, E if

MET,77ED"F

71(1,M)ET.

Theorem 0.14 (Minimax Principle). If T is a family of subsets of W,

c:= inf sup 'D(u) MET ueM

is finite, T is invariant under D,,E for all sufficiently small e > 0, and 4D satisfies (C)°, then c is a critical value of 4D. We say that a family F of continuous maps ry from a topological space X into W is invariant under D,,F if

''EP,?7ED, Minimax principle is often applied in the following form, which follows by taking T = {y(X) : -y e P} in Theorem 0.14.

Theorem 0.15. If P is a family of continuous maps ry from a topological space X into W, c:= inf

sup

'F(u)

'7Er nE7(X)

is finite, P is invariant under D°E for all sufficiently small e > 0, and (D satisfies (C),, then c is a critical value of OD.

Some references for Theorems 0.14 and 0.15 are Palais [93], Nirenberg [89], Rabinowitz [111], and Ghoussoub [53]. Minimax methods were introduced in Morse theory by Marino and Prodi [79]. The following result is due to Liu [71].

Theorem 0.16. If o- E Hk(4, V) is a nontrivial singular homology class where -co < a < b < +oo are regular values, c := inf sup 'F(u) zEa telzl

where zj denotes the support of the singular chain z, 'F satisfies (C)°, and

K° is a finite set, then there is a u e K° with Ck('F, u) # 0.

8

0. MORSE THEORY AND VARIATIONAL PROBLEMS

0.5. Linking The notion of homotopical linking is useful for obtaining critical points via the minimax principle.

Definition 0.17. Let A be a closed proper subset of a topological space X, g e C(A,W) such that g(A) is closed, B a nonempty closed subset of W such that dist(g(A), B) > 0, and r = {ry c C(X, W) : 7(X) is closed, ryI A = g}. We say that (A, g) homotopically links B with respect to X if

7(X)nB#0 V'yeI'. When g : A c W is the inclusion and X = to : it c A, t e [0, 1] }, we simply say that A homotopically links B. Some standard examples of homotopical linking are the following.

Example 0.18. If no e W, U is a bounded neighborhood of no, and ul 0 U, then A = {uo,uul} homotopically links B = OU. Example 0.19. If W = W1 Q+ W2,, u = ul +u2 is a direct sum decomposition of W with WI nontrivial and finite dimensional, then A = {ui C WI : Ilul II = R} homotopically links B = W2 for any R > 0.

Example 0.20. If W = WI(DW2i it = ul+u2 is a direct sum decomposition with WI finite dimensional and v e W2 with IIvhl = 1, then A = {ul e WI : Iul1I S R} u {u = ur + tv : ul e W1, t > 0, IIuhI = R} homotopically links

B = {u2eW2:IIu2II=r} for any00, Ck+1(4', nl) # 0. Theorems 0.30 and 0.31 and Corollary 0.32 also hold under the more general notion of homological local linking introduced in Perera [96].

0.7. p-Laplacian The p-Laplacian operator Op u = div (IV u VP-2 VU),

P

C (1, CO)

arises in non-Newtonian fluid flows, turbulent filtration in porous media, plasticity theory, rheology, glacelogy, and in many other application areas; see, e.g., Esteban and Vazquez [48] and Padial, Takac, and Tello [90]. Problems involving the p-Laplacian have been studied extensively in the literature

during the last fifty years. In this section we present a result on nontrivial critical groups associated with the p-Laplacian obtained in Perera [98]; see also Dancer and Perera [40]. Consider the nonlinear eigenvalue problem

-OPu=,AlulP-2u in Q (0.2)

l

u=0

on51

12

0. MORSE THEORY AND VARIATIONAL PROBLEMS

where t2 is a bounded domain in R°`, n >, 1 and Apu = div (IVulp-2 Vu) is the p-Laplacian of it, p e (1, oc). It is known that the first eigenvalue AI is positive, simple, has an associated eigenfunction that is positive in 12, and is isolated in the spectrum o(-Ap); see Anane [9] and Lindgvist [68, 69]. So the second eigenvalue A2 = inf o-(-Op) n (A1, m) is also defined; see Anane and Tsouli [8]. In the ODE case n = 1, where 12 is an interval, the spectrum consists of a sequence of simple eigenvalues Ak / cc and the eigenfunction associated with Ak has exactly k-1 interior zeroes; see Cuesta [35] or Drabek [46]. In the semilinear PDE case it >, 2, p = 2 also o-(-A) consists of a sequence of eigenvalues Ak J' cc, but in the quasilinear PDE case it 3 2, p # 2 a complete description of the spectrum is not available. Eigenvalues of (0.2) are the critical values of the C'-functional

I(u) = f Vulp,

u e S = {u e W = Wo'p(52) : iu LP(s) = 1},

which satisfies (PS). Denote by A the class of closed symmetric subsets of S and by ry+(A) = sup {k 3 1 3 an odd continuous map Sk-1 Al,

-y-(A) = inf {k > 1 : 3' an odd continuous map A -*

Sk-1}

the genus and the cogenus of A e A, respectively, where Sk-1 is the unit sphere in Rk. Then A-. k =

inf

AEA

sup I(u),

k > 1

uEA ry±(A)>-k

are two increasing and unbounded sequences of eigenvalues, but, in general,

it is not known whether either sequence is a complete list. The sequence (A+) was introduced by Drabek and Robinson [47]; yk is also called the Krasnosel'skii genus [60]. Solutions of (0.2) are the critical points of the functional I,\ (u) = f I VujP _ A IuIp,

u c-

W,pQ.

When A 0 a(-op), the origin is the only critical point of Ia and hence the critical groups Cq(Ia,0) are defined. Again we take the coefficient group to be Z2. The following theorem is our main result on them.

Theorem 0.33 ([98, Proposition 1.1]). The spectrum of -Op contains a sequence of eigenvalues \k / co such that A- < Ak < A+ and A E (Ak, Ak+1)\a(-Ap)

:

Ck(IA, 0) : 0.

Various applications of this sequence of eigenvalues can be found in Per-

era [99, 100], Liu and Li [75], Perera and Szulkin [105], Cingolani and Degiovanni [30], Guo and Liu [55], Degiovanni and Lancelotti [43, 44], Tanaka [129], Fang and Liu [50], Medeiros and Perera [82], Motreanu and Perera [86], and Degiovanni, Lancelotti, and Perera [42].

0.7. p-LAPLACIAN

13

The eigenvalues Ak are defined using the Yang index, whose definition and some properties we now recall. Yang [133] considered compact Hausdorff spaces with fixed-point-free continuous involutions and used the Cech homology theory, but for our purposes here it suffices to work with closed symmetric subsets of Banach spaces that do not contain the origin and singular homology groups. Following [133], we first construct a special homology theory defined on the category of all pairs of closed symmetric subsets of Banach spaces that do not contain the origin and all continuous odd maps of such pairs. Let (X, A), A c X be such a pair and C(X, A) its singular chain complex with Z2-coefficients, and denote by T# the chain map of C(X, A) induced

by the antipodal map T(u) = -u. We say that a q-chain c is symmetric if T#(c) = c, which holds if and only if c = c' + T#(c') for some q-chain c'. The symmetric q-chains form a subgroup Cq(X,A;T) of Cq(X,A), and the boundary operator o"q maps Cq(X, A; T) into Cq_1(X,A;T), so these subgroups form a subcomplex C(X, A; T). We denote by Zq(X, A; T) = {c e Cq(X, A; T) : dqc = 01, Bq(X, A; T) = {8q+1c : c c Cq+1(X, A; T)},

Hq(X,A;T) = Zq(X,A;T)/Bq(X,A;T) the corresponding cycles, boundaries, and homology groups. A continuous odd map f (X, A) -> (Y, B) of pairs as above induces a chain map f# C(X,A;T) C(Y,B;T) and hence homomorphisms

f.:Hq(X,A;T) For example,

Hq(Y,B;T).

7G2,

0.gSk

0,

q>k+1

Hq (9k ;T) =

(see [133, Example 1.8]).

Let X be as above, and define homomorphisms v : Zq(X;T)

7L2

inductively by In (c),

q = 0

v(ac),

q>0

v(z) _

nq o, is defined by if z = c + T# (c), where the index of a 0-chain c = In(c) = L n,. As in [133], v is well-defined and vBq(X;T) = 0, so we can define the index homomorphism v, : Hk(X;T) - Z2 by v.([z]) = v(z). If F is a closed subset of X such that F u T(F) = X and A = F nT(F), then

there is a homomorphism

A: Hk(X;T) -* Hq-1(A;T) such that v.(A[z]) = v.([z]) (see [133, Proposition 2.8]). Taking F = X we see that if v. Hk(X;T) = Z22i then v. Hq(X;T) = Z2 for 0 , -1 : v Hk+l(X;T) = 01,

taking info = oo. Clearly, v.Ho(X;T) = Z2 if X # 0, so i(X) = -1 if and only if X = 0. For example, i(Sk) = k (see [133, Example 3.4]).

Proposition 0.34 ([133, Proposition 2.4]). If f : X -> Y is as above, then v.(f,([z])) = v.([z]) for [z] e Hq(X;T), and hence i(X) < i(Y). In particular, this inequality holds if X c Y.

Thus, k+-1 < i(X) < k--1 if there are odd continuous maps Sk+-1

X

so

'y+ (X) < i(X) + 1 < -Y-(X).

(0.3)

Proposition 0.35 ([98, Proposition 2.6]). If i(X) = k > 0, then Hk(X) # 0.

PROOF. We have

7G2, 0

>

so it follows from the exact sequence of the pair (I° n U, I° n U\ {0}) that Hq(I° n U, I° n U\ {0})

Hq_1(I° n S).

Since 1),Is =I-A,I°nS=IA. We are now ready to prove Theorem 0.33.

0.7. pLAPLACIAN

15

PROOF OF THEOREM 0.33. Set Ak =

inf

AEA

sup I(u),

k , 1.

uEA

i(A)>k-1

Then (Ak) is an increasing sequence of critical points of I, and hence eigen-

values of -AP, by a standard deformation argument (see [98, Proposition 3.1]). By (0.3), Ak < Ak < Ak , in particular, Ak co. Let A E (Ak, Ak+l)\Q(-Op). By Lemma 0.36, Ck(Ix, 0) .: Hk_1(I'), and Ia e A since I is even. Since A > Ak, there is an A e A with i(A) > k - 1

such that I < A on A. Then A c Ia and hence i(I") > i(A) > k - 1 by Proposition 0.34. On the other hand,i(I') < k - 1 since I < A c Ak+1 on P. So i(IA) = k - 1 and hence Hk_1(IA) 0 0 by Proposition 0.35.

CHAPTER I

Abstract Formulation and Examples Recall that an operator q : W is s-homogeneous, where s > 0, if

W', from one Banach space to another,

q(au) = as q(u)

Vu E W, a

0.

Taking a = 0 gives q(0) = 0. Note that q is s-homogeneous and odd if and only if q(au) = lals-1 a q(u) Vu E W, a c and s-homogeneous and even if and only if q(au) = I ajs q(u)

Vu e W, a E R.

A continuous homogeneous operator is bounded in the following sense.

Proposition 1.1. If q e C(W,W') is s-homogeneous, then there is a constant C > 0 such that Ilq(u)ll js+1 llujli9

Let uj = ui/(j llu7p). Then 11u;p =1/j llq(n;)II =

llg(u,)il

Vj.

0, but

> j - Go

a llujll

contradicting the continuity of q. Let L(W, W') be the Banach space of bounded linear operators from W to W'. Then q : U W', where U is an open set in W, is differentiable at it e U if there is a q'(u) e L(W, W'), called the derivative of q at it, such

that

q(u+v)=q(u)+q'(u)v+o(Bvll) as

in W.

The map q is continuously differentiable on U, or belongs to the class C1(U,W'), if q' E C(U, L(W, W')), i.e., q' is defined everywhere in U and U -. L(W, W'), u q'(u) is continuous. 17

1. ABSTRACT FORMULATION AND EXAMPLES

18

Let W* be the dual of W and the pairing between W* and W. Then q e C(W, W*) is a potential operator if there is a functional Q e C' (W, R), called a potential for q, such that

Q'(u) = q(u) Vu e W. Replacing Q with Q - Q(O) if necessary, we may assume that Q(O) = 0.

Proposition 1.2. If q is a potential operator and Q is its potential satisfying Q(0) = 0, then

Q(u) =

Cl

(q(tu), u) dt Hu e W.

0

In particular, Q is even if q is odd. If q is s-homogeneous, then

Q(u) = 5 + 1

Hue W

(q (u), u)

and is (s + 1)-homogeneous. PROOF. We have

fl d (Q(tu)) dt r 1 (Q'(tu),u) dt = (q(tu),u) dt. Q(u) = o dt f1 = Jo 0 The last integral equals fl

is (q(u), u) dt = 0

1

s+1

(q (u), u)

when q is s-homogeneous.

We consider the nonlinear operator equation (1.1)

in the dual (W*,

d,1cdcoo, where

Apu= f(u) of a real reflexive Banach space (W, I ) of dimension is

(A,) (p - 1)-homogeneous and odd for some p e (1, oo), (A2) uniformly positive: 3 co > 0 such that (Ap u, u) > co Ilujlp

Vu e W,

(A3) a potential operator, (A4) of type (S): every sequence (uj) c W such that

u.-u, (Apuj,%-u)- 0 has a subsequence that converges strongly to u, and f : W -> W* is a compact potential operator. By Proposition 1.2, the potential Ip of Ap satisfying Ip(O) = 0 is given by (1.2)

I4(u) = I (Apu,u)

1. ABSTRACT FORMULATION AND EXAMPLES

19

and is p-homogeneous and even, and the potential F off satisfying F(0) = 0 is given by

F(u) =

(1.3)

1

(f (tu), u) dt.

0

Let (1.4)

lb(u) = Ip(u) - F(u) = p (Ap u, u) -

Jo

(f (to), u) dt,

u e W.

Then V(u) = 1(u) - F'(u) = Ap u - f(u),

so the solutions of equation (1.1) coincide with the critical points of the C1-functional 4D, i.e., u e W solves (1.1) if and only if V(u) = 0. We note that 4D(0) = 0, and (D is even when f is odd. By Proposition 1.1, there is a constant Co > 0 such that IIApull* ' C0 Dull -1

(1.5)

du e W.

Combining (1.2), (A2), and (1.5) gives (1.6)

P

Vu E W.

IIuIIp (II(u) s Po IIuIIp

In particular, Ip ' 0 and is bounded on bounded sets. Since the compact operator f maps bounded sets into precompact, and hence bounded, sets, F is also bounded on bounded sets by (1.3).

Therefore 4 is bounded on

bounded subsets of W. In many important applications (1.7)

(Apu,v) 0, or this can be achieved with respect to an equivalent norm.

Proposition 1.3. If W is uniformly convex and (1.7) holds, then (A4) is satisfied.

PROOF. If uj (1.8)

u and (Ap uj, uj - u)

0, then

0'< r(Ilujllp-l-IIuIIp-1)(IIujII-(lull) (Ap uj, uj) - (Ap uj, u) - (Ap u, uj) + (Ap u, u)

_ (Ap uj, uj - u) - (Ap u, uj - u) so ll uj ll - Hull and hence uj

0,

u by uniform convexity.

Next we give some examples. In all the following examples (A1) - (A3) are clear and (A4) follows from Proposition 1.3 and Holder inequalities.

1. ABSTRACT FORMULATION AND EXAMPLES

20

1.1. p-Laplacian Problems Our motivating example is the p-Laplacian problem with the Dirichlet boundary condition

Opu= f(x,u) inQ

u=0

ona 52

where 0 is a bounded domain in R, n >, 1, Ap u = div (I VUlp-2 Vu) is the p-Laplacian of it, p e (1, co), and the nonlinearity f is in the space Car(Q x R) of Caratheodory functions on Q x R and has subcritical growth, i.e., (1.10)

if (x, t)l S C

+ 1)

(ltIT-1

V(x, t) E 0 X R

for some r e (1, p*), where nP P*

=

n-p

P 0. Here W is the Sobolev spacd Wd'p(Q) with the usual norm

f

1

(lull = (J IDulp)

p

and the dual W-l,y'(Q) where p' = p/(p - 1) is the Holder conjugate of p, lpulp-2 Vu - Vv,

(Ap it, v) = fn

II(u) =

p f lVulp,

and (1.11)

(f(u),v) = f f(x,u)v,

F(u) =

n

f sz

F(x,u)

where c

(1.12)

F(x, t) =

f (x, s) ds. J0

The operator f is compact by the growth condition (1.10) and the compactness of the Sobolev imbedding WJ" (Q) L'(Q) for r < p*. We will use this example throughout the text to illustrate our results for equation (1.1). In the following sections we give some generalizations.

1.2. AP Laplacian Problems Problem (1.9) is the special case ai1(x) = Sij of the Ar Laplacian problem Ap u = f (x, u) in 52 U=0

on d 0

1.3. PROBLEMS IN WEIGHTED SOBOLE\' SPACES

21

where p_2

auau)2

(n

Apu=

akl (x) k,1=1

= div

(IVU11-

A(x2)

au

aj(x) ay;

A(x)Vu)

is the AP Laplacian of u, each a satisfies

E, axl J

= anti E L'(Q), and the matrix A = (aij)

aA(x)

12 H E I

a.e.

Now we use the norm 1

P

hull = (J IVuIA(x) in W01"(Q) and take

f

(Ap u, v) = o I VulA(x) A(x)Vu Vv,

II(u) =

f

A151.

p Jn

2

We note that the special case a;j (x) a(x)p b;j, 1 , 1 since s >, 1/(p - 1). So

(1.14)

W

Wol"(Q) " Lr(cl)

I

UCW

1. ABSTRACT FORMULATION AND EXAMPLES

22 for

n.s

+

P S -

r

Ps < n ns - (ps - n)'

ps>n, (Y, B) of pairs is a map

f: X -. Y such that f (A) c B. Two maps of pairs fo, fl : (X, A) -* (Y, B) are homotopic, written fl, if there is a map F X x [0, 1] Y, called a homotopy between fo and fl, such that (i) F(, 0) = fo, fo

(ii) F(, t) : (X, A) (iii) F(.,1) = fl.

(Y, B) is a map of pairs for all t e [0,1],

F is called a homotopy relative to A if F(-, t) IA is independent of t. The homotopy class of f : (X, A) -. (Y, B), denoted by [f], is the set of maps homotopic to f. The set of all homotopy classes of maps from (X, A) to (Y, B) is denoted by [(X, A), (Y, B)].

Two pairs of spaces (X, A) and (Y, B) are homotopy equivalent, or have the same homotopy type, written (X, A) ti (Y, B), if there are maps f : (X, A) --. (Y, B) and g : (Y, B) (X, A), called homotopy equivalences between (X, A) and (Y, B), such that fg

id(Y,B),

gf

id(x,A).

A retraction of X onto A is a map r : X

A such that rIA = idA.

A deformation retraction of X onto A is a homotopy relative to A between

idX and a retraction r : X -, A, i.e., a map F : X x [0, 1] - X such that (i) F(-, 0) = idx, (ii) F(., t)IA = idA Ht E [0, 1], (iii) F(X, 1) = A. Then A is called a deformation retract of X, and since ri = idA and it id X where i : A c X is the inclusion, X A. A space is contractible if it has the homotopy type of a one-point space, in particular, if it deformation retracts to a point.

Example 2.1. RTh, D', and S' are all contractible. Triples (X, A, B) of spaces X D A D B give rise to similar homotopy notions.

2.2. DIRECT LIMITS

29

2.2. Direct Limits We briefly recall the notions of directed systems and their limits. A partial order on a set A is a relation < in A that is (i) reflexive: A 5 A VA e A,

(ii) transitive: A < p < v

A < v. A partially ordered set A is directed if for each pair A, p e A, there is a v e A

with A 0 when X is a metric space, so we assume that k < oo.

The set A of all neighborhoods of A in X is directed downward by inclusion. If U is a neighborhood of A, there is a closed neighborhood

V c U of A since X is normal and A is closed, and then V n -V c U is a closed invariant neighborhood of A since A is invariant, so the subset N of closed invariant neighborhoods Na is cofinal in A. Thus, by Proposition 2.7,

j* = lim7a : lmHk(N),) ^ Hk(A) N N where ja : A c NA. Now, we have the commutative diagram -* aA

Hk(N,)

N

f* Hk (RPCO)

lim H (N) A

(fAA )*

Hk(A)

where fA is the classifying map of Na, with faj,, serving as a classifying map for A. Since i(A) = k, zafa(wk) _ (fxaa)*(wk) = 0,

and since j* is an isomorphism, then iafa(wk) = 0. Proposition 2.2 now gives a p e N such that (fa2Aµ)*(wk) = 2AµfA(wk) = 0

2.5. COHOMOLOGICAL INDEX

39

where iaµ : Nµ c NA, with faixN, serving as a classifying map for Nµ, so i(N,1) < k. The desired conclusion follows since k i(N,1) by (i2). When X is a metric space and A is compact, b := dist(A, Nµ) > 0, and i(N5(A)) = k since A c Na(A) c NN,. (i5) Clearly, we may assume that k := i(A) and l := i(B) are finite. By (i,I), there are closed invariant neighborhoods M of A and N of B in X such

that i(M) = k,

i(N) = 1,

and we have the commutative diagram

Hk(X, M)

A*

I Hk(X)

f*

'

Hk(M)

(fo*

Hk(RPc) where the top row comes from the exact sequence of the pair (X, M) and f is the classifying map of X, with f i I serving as a classifying map for M. Since i(M) = k, iif*(wk) _ (fiI)*(wk) = 0,

so f*(wk) e keril = j1(Hk(X,M)) by exactness, say, f*(wk) = jra1. Similarly, f *(wt) = j2 a2 for some a2 E HI (X, N) where j2 : X C (X, N). Now, since the interiors of M and N cover X, we have the commutative diagram Hk+l(X, M u N) Hk(X, M) Ix HI (X, N)

i xi

i*1 H"(X) x HI(X) Hk+1(X) where the horizontal maps are cup products and j c (X, M u N). Since 1

MuN=X and hence H*(X,Mu N) = 0, f*(wk+l)

= f *(wk " WI) = f*(wk) .. f*(WI)

al "j2 a2 =i*(al' a2)=i*0=0, *

= .71

soi(X) i(B). We will show that the map f = Pr1I.: B -> A, where Pr1 : Y A is the projection onto the first factor, induces injections f* : H"(A) HQ(B) for all q and hence i(B) > i(A). Since the retraction r = Pr1lyo Yo A induces injections r*, it suffices to show that io : B c Yo induces injections io.

f

H°(A) r H(Yo) H0(B)

We have the inclusions Yo io

A x {0}

ko

to

7o

B

Y Y1

k

Ax{1}

and the Mayer-Vietoris exact sequence

Hq(y) '- HQ(Y0)@H,(Y1)

`2

HQ(B)

where j* = (j0, -jr) and P = io + ii (see, e.g., Spanier [1261). If ioa = 0,

*(a,0)=ioa+210=0, so (a,0) e keri* = j*(H4(Y)) by exactness, say, (a,0) =.I *f3 = (jof3,-310). Then jif = 0 and hence 110 = kiji/3 = 0, which implies that ,Q = 0 since l1, and hence also 11, is a homotopy equivalence, so a = joQ = 0. (is) Taking R > 0 so large that BR(0) D U and applying Corollary 2.13 below (whose proof uses only (i2) and (i7)) to the identity map on ISR(O) = BR(0) gives

i(W) = i(BR(0) n aU) > i(SR(0)) = i(S) = dimes, and (i3) gives the opposite inequality.

2.5. COHOMOLOGICAL INDEX

41

Corollary 2.13. If U is a bounded closed symmetric neighborhood of the origin in a nonmed linear space W, A is a bounded symmetric subset of U, and 4 : IA W is an odd continuous mapping such that vl)(IA) is closed and ?IA = idA, then i(v/)(IA) n SU) > i(A). PROOF. We have z/ (IA) n SU = zl (b-1(aU)), so

i(v/5(IA) n 8U) > i(r-1(5U)) by (i2). Since ?G is odd, v/b(0) = 0, so there is a 6 > 0 such that O

V6:= {tx : x e A, t E [0, 6)} c v)-'(U) by continuity. We apply (i7) to o : A x [0,1] IA\V4 = (vG-1(U)\Vs) u (2.3)

(x,t)'-* ((1 -t)6+t)x. Since V5 is contained in the closed set ip-1(U), so is its relative boundary dV4, so yp(A x {0}) = 5714 c v,-1(U)\V4. Since v/' is the identity on A c U',

o(A x {1}) = A c 0-1(U). Since p is onto, w(A x [0,1]) n (,G-1(U)\V5) n 0-1(U) _-1(5U)\14 = v/ -'(5U) by (2.3), so we have i(v/)-1(57U)) > i(A).

The following proposition gives some additional properties of the index.

Part (iii) is due to Degiovanni and Lancelotti [43], and (iv) to Cingolani and Degiovanni [30].

Proposition 2.14. Let X E .F with index k > 1. (i) If X is the disjoint union of a pair of subsets U, -U, then k = 1.

In particular, k = 1 when X is a finite set. (ii) If X is compact, then k < oo.

(iii) If X is locally contractible, then for each finite j < k, X has a compact invariant subset C with i(C) >, j. In particular, there is a compact invariant subset C with index k when k < oo. (iv) If k < oo and A is an invariant subset of X with index k, then the rank of i* : Hk-1(X) , Hi`-1(A), induced by i : A c X, is at least 1 + 6k I. In particular, Hk-1(X) # 0. PROOF. (i) The map

f:X-S°,

x--.

1,

-1, x E -U

is equivariant, so

1(Ic i(S°)=1 by (i2) and Example 2.11.

xEU

2. BACKGROUND MATERIAL

42

(ii) Each x e X has a closed neighborhood U. such that Ux n -Ux = 0 since x # -x and X is regular, and a finite number of them, Uxl, ... , U, cover X by compactness. Then m

i(X) (L, u), in particular, (Lj, uj - u) 0. PROOF. Since (uj) is bounded by the principle of uniform boundedness,

I(Lj,uj)-(L,u)Is1ILj-LII*11ujll+I(L,uj)-(L,u)I-*0.

3.2. DEFORMATION LEMMAS

47

3.2. Deformation Lemmas Deformation lemmas allow to lower sublevel sets of a functional, away from its critical set, and are an essential tool for locating critical points. The main ingredient in their proofs is usually a suitable negative pseudo-gradient flow, a notion due to Palais [93].

Definition 3.5. A pseudo-gradient vector field for 4) on W is a locally Lipschitz continuous mapping V : W W satisfying

V(v)

(3.1)

(114)f(u)11*)2

V(u)2 (V(u), V(u))

Vu C W.

Lemma 3.6. There is a pseudo-gradient vector field V for 4' on W. When 4' is even, V may be chosen to be odd. PROOF. For each u e W, there is a w(u) e W satisfying IIw(u)II < 11V(u)11*

,

2 (4 (u), w(u)) > (b (u)II*)2

by the definition of the norm in W*. Since 4V is continuous, then (3.2)

hw(u)11 _ D

(v)11* ,

2 (4'(v),w(u)) > (l p,(v)11*)2

Vv e Nu

for some open neighborhood Nu c W of it. Since W is a metric space and hence paracompact, the open covering {Nu}uEw has a locally finite refinement, i.e., an open covering {Na},EA of W such that (i) each Na c Nua for some ua e W, (ii) each u e W has a neighborhood U. that intersects Na only for A in some finite subset A. of A (see, e.g., Kelley [59]). Let {cpa},,eA be a Lipschitz continuous partition of unity subordinate to {Na},EA, i.e., (i) spa e Lip (W, [0,1]) vanishes outside Na, (ii) for each it e W, Z cpa(u) = 1

(3.3)

.SEA

where the sum is actually over a subset of Au, for example, dist(u, W\Na)

a

(u)

' dist (u,

W\N,).

AeA

Now V(u)=

' Wa(u)w(ua) AeA

is Lipschitz in each Uu and satisfies (3.1) by (3.2) and (3.3).

3. CRITICAL POINT THEORY

48

When $ is even, V is odd and hence -V(-u) is also a pseudo-gradient, and therefore so is the odd convex combination (V(u) - V(-u)). The following deformation lemma improves that of Cerami [25].

Lemma 3.7 (First Deformation Lemma). If c e 1@, C is a bounded set containing ICS, 5,k > 0, and 4) satisfies (C)0, then there are an eo > 0 and,

for each e e (0,so), a map g e C([0,1] x W, W) satisfying

(i) q(0,) = idw, (ii) rl(t,) is a homeomorphism of W for all t e [0, 1], (iii) ?7(t,) is the identity outside A = (0, u), but ((tj,uj) -» ((0,u) = u. Then there is a 6 > 0 such that (B36(u)\ {u}) n K° _ 0 and

II((tj,uj)-ull X26

(3.19)

for a subsequence. Since ((0, uj) = uj -* u,

II((0,uj) - ull < 6

(3.20)

for sufficiently large j. By (3.19) and (3.20), uj E 4pb\(,pd u K6), tj > 0, and there are sequences 0 < s' < tj < tj for which (3.17), and hence also (3.18), holds. Case 4: u. e

t = 0. Then ((0, u) = u.

3.3. Minimax Principle First deformation lemma implies that if c is a regular value and (P satisfies (C)C, then the family D, E of maps 77 E C([0, 1] x TV, W) satisfying (i) 77(0, .) = idw,

(ii) r7(t, ) is a homeomorphism of W for all t c [0,1], (iii) 77(t,) is the identity outside DI+26 for all t e [0,1], (iv) 4 u)) is nonincreasing for all u e W, (v)

77(1,

Dc+e)

C P`

is nonempty for all sufficiently small e > 0. We say that a family F of subsets of W is invariant under E),,, if

MaT,77ED,,E

s7(1,M)EF.

Proposition 3.10 (Minimax Principle). If F is a family of subsets of W. (3.21)

c := inf sup (D(u) MeT nEAt

is finite. F is invariant under DC,e for all sufficiently small e > 0, and 4" satisfies (C)C, then c is a critical value of D. PROOF. If not, taking e > 0 sufficiently small, M E F with sup 4 (M) < c+ e, and 7) E D..E, we have r7(1, M) E F and sup q )(77(l, M)) < c - e by (v), contradicting (3.21).

3.4. CRITICAL GROUPS

53

We say that a family r of continuous maps 7 from a topological space X into W is invariant under D0, E if

yEP,rleD,,E

q(i,-)o-Ycr.

Minimax principle is often applied in the following form.

Proposition 3.11 (Minimax Principle). If P is a family of continuous maps y from a topological space X into W, c:= inf sup 4)(u) ryeP uE'Y(X)

is finite, r is invariant under I3. e for all sufficiently small e > 0, and 4) satisfies (C)c, then c is a critical value of D. PROOF. Apply Proposition 3.10 with .F = {y(X) : -Y C- 171,

0

3.4. Critical Groups In Morse theory the local behavior of D near an isolated critical point u is described by the sequence of critical groups Cq((p, u) := Hq((D° n U, 4)` n U\ {u}), q > 0 (3.22) where c = 4)(u) is the corresponding critical value and U is a neighborhood of u containing no other critical points of 4). They are independent of U, and hence well-defined, by (c3). Critical groups help distinguish between different types of critical points and are extremely useful for obtaining multiple critical points of a functional (see, e.g., Chang [28)). One of the consequences of the second deformation lemma is the following proposition relating the change in the topology of sublevel sets across a critical level to the critical groups of the critical points at that level.

Proposition 3.12. If -co < a < b < +cc and 4) has only a finite number of critical points at the level c E (a, b), has no other critical values in [a, b], and satisfies (C)e for all c' E [a, b] n R, then

Hq(,D6,,1)a). O Cq(4),u)

Vq.

ueK0

In particular,

dimHq(4)b,(ba) =

)' dimCq(4),u)

Vq.

uEK°

PROOF. We have Hq(4)b, (Da) , Hq(4)`,'Fa) x Hq(4)`, 4)'\K`) (3.23)

since 4)` and 4)d are deformation retracts of 4)b and 4)`\K°, respectively, by Lemma 3.9. Taking 6 > 0 so small that the balls B5(u), U E K` are mutually disjoint and then excising D`\ UuEK° B5(u), we see that the last group in (3.23) is isomorphic to O Hq(4)` n B5(u), 4)` n BS(u)\ {u}) = O Cq(4), u). uEK0

uEK.

3. CRITICAL POINT THEORY

54

For the change in the topology across multiple critical levels, we have

Proposition 3.13. If -oo < a < b < +co are regular values and 4' has only a finite number of critical points in oba and satisfies (C), for all c e [a, b] n R, then

dimH9(4)b,.pa) < )' dimC4($,u)

Vq.

uEKa

In particular, $ has a critical point it with a < $(u) < b and C'(4), u) # 0 when HQ(4b, (pa) # 0.

First we prove a lemma of a purely topological nature. Lemma 3.14. If X1 c ... c Xk+l are topological spaces, then k

dimH9(Xi+1,Xi)

dimH4(Xk+1,X1)
0. PROOF. If not, there is a ry e C(IA, BC) with 71A = id _q. Then SAO

M\Bo,

JrM (-y ((I - t) Ru)), (u, t)

t e [0, 1]

-r,I,I(ry(-(1+t)Ru)), te[-1,0)

is an odd map and hence i(M\Bo) i(SAo) = i(Ao) + 1 by (12) and (i6), contradicting (3.28).

Proposition 3.23. If h e C(CAO, M) is such that h(CAo) is closed in M and hIA0 = idA0, then A = R(IAo u h(CAo)) homotopically links B = rBo

for any0(u) > a and Cq+1(4 u) # 0. PROOF. We have the commutative diagram

Hq(B`)

Hq(4ia)

Hq(A)

induced by A c r c B`. Since i* # 0, Hq(4?Q) # 0 and hence the conclusions follow from Propositions 2.4 (ii) and 3.16.

To construct an example, let M, 7rM, As, and B° be as in the last section.

Proposition 3.26. Then A = RA° cohomologically links B = irk (Bo) u{0} in dimension q = i(Ao) - 1 for any R > 0. PROOF. We have the commutative diagram

Hq(M\Bo)

Hq(B`)

Hq(Ao)

''

Hq(A)

3.8. NONTRIVIAL CRITICAL POINTS

59

where j : A0 c M\B0 and the vertical maps are isomorphisms induced by M\Bo and the homeomorphism the homotopy equivalence rrM16 : B' nM I A : A -* A0. The conclusion follows since j * # 0 by Proposition 2.14 (iv).

If A cohomologically links B in dimension q and h is a homeomorphism of W, then the following commutative diagram, where j : h(A) c h(B`), shows that h(A) cohomologically links h(B) in dimension q as well. H9(hI(B`))

'*

h* I

HQ(h (A)) h*

Hq(LLLB')

Hq(A) In particular, we have the following generalization of Proposition 3.26.

Proposition 3.27. If h is a homeomorphism of W, then A = h(RAo) cohomologically links B = h(7r - (Bo) u {0}) in dimension q = i(Ao) - 1 for

any R>0.

3.8. Nontrivial Critical Points In many applications D has the trivial critical point u = 0 and we are interested in finding others. We assume that 4) has only a finite number of critical points. The following proposition is useful for obtaining a nontrivial critical point with a nontrivial critical group. Proposition 3.28. Assume that 4? satisfies (C). (i) If Cq(4?, 0) = 0 and C"(-D, oo) # 0 for some q, then ID has a critical point It # 0 with Cq((D, u) # 0. (ii) If Cq(4., 0) # 0 and Cq(4i, oo) = 0 for some q, then 4i has a critical point It # 0 with either 4?(u) < 0 and Cq-1(4?, u) # 0, or 4i(u) > 0 and Cq+'(,Du) # 0. First a purely topological lemma. Lemma 3.29. If X1 c X2 c X3 c X4 are topological spaces, then dimHq-1(X2iXI)+dimHq+1(X4,X3)

dimHq(X3iX2)-dimHq(X4iX1)

Hq.

PROOF. From the exact sequence

...

Hq-1(X2, XI)

HQ(X3, X2)

HI (X3, Xj) of the triple (X3, X2, X1), we have

dim Hq-1(X2, X1) > rank 6 = nullity j * = dim Hq(X3, X2) - rankj* dim Hq(X3, X2) - dim Hq(X3i XI),

3. CRITICAL POINT THEORY

60

and from the exact sequence Hq(X4, X1)

Hq(X3, XI) a

H4+1(X4 X3)

...

of the triple (X4, X3, XI), dim Hq(X3i XI) = rank 6+ nullity b < dim Hq+I (X4, X3) + rank i* < dim Hq+I(X4, X3) + dim Hq(X4, X1), so the conclusion follows. PROOF OF PROPOSITION 3.28. (i) By Proposition 3.16, D has a critical point u with Cq(4), u) # 0 since Cq(4), 00) # 0, and u # 0 since C"(4), 0) = 0.

(ii) Let e > 0 be so small that zero is the only critical value in [-e, e]

and a be less than -e and all critical values. Since dimHq((DE 4)-E) dimCq(4), 0) by Proposition 3.12 and Hq(W, 4)n) = Cq(4), co), applying Lemma 3.29 to 4a c $-E c V c W gives dim Hq-I ((I)-E, 4)")+ dim Hq+1 (W, 4p6)

dim Cq(1, 0) - dim Cq(4), oo) > 0.

Then either

Hq-1($-E 41) # 0,

or Hq+I(W, V)

0, and the conclusion

follows from Proposition 3.13.

Remark 3.30. The alternative in Proposition 3.28 (ii) and Lemma 3.29 were proved by Perera [95, 96].

3.9. Mountain Pass Points A critical point it of 4) with C' (4), u) # 0 is called a mountain pass point. Since cohomology groups, and hence also critical groups, are trivial in negative dimensions, the special case q = 0 of Proposition 3.28 (ii) reduces to

Corollary 3.31. If C°((D, 0) # 0, C°(4), oo) = 0, and 4) satisfies (C), then 4) has a mountain pass point it # 0 with D(u) > 0. This implies the well-known mountain pass lemma of Ambrosetti and Rabinowitz [7]. Indeed, if the origin is a local minimizer and 4) is unbounded

from below, then C°(4), 0) x Z2 by Proposition 3.18 and C°(4), co) = 0 by Proposition 3.15 (ii), so Corollary 3.31 gives a positive mountain pass level.

3.10. Three Critical Points Theorem Another consequence of Proposition 3.28 (ii) is

Corollary 3.32. If Cq(4), 0) # 0 for some q > 1 and 4) is bounded from below and satisfies (C), then D has a critical point ui # 0. If q >, 2, then there is a second critical point u2 # 0.

3.11. COHOMOLOGICAL LOCAL SPLITTING

61

PROOF. By Corollary 3.19, Cq(,D, co) = 0 and D has a global minimizer

u1 with Cq(*F, ul) = 0. Since C" ((P, 0) # 0, u1 # 0 and there is a critical point u2 # 0 with either Cq-1('F,u2) # 0 or Cq+1(4,u2) # 0. When q ? 2, u2 # u1 since Cq-1('F,u1) = Cq+I('F,u1) = 0. 0

3.11. Cohomological Local Splitting The notion of cohomological local splitting is useful for obtaining nontrivial critical groups at zero and hence nontrivial critical points via Proposition 3.28 (ii).

Definition 3.33. We say that 'F has a cohomological local splitting near zero in dimension q, 1 < q < co if there are (i) a bounded symmetric subset M of W\ {0} that is radially homeomorphic to the unit sphere, and disjoint symmetric subsets A0

and B0 of M such that

0

i(Ao) = i(M\Bo) = q,

(3.32)

(ii) a homeomorphism h from IM onto a neighborhood U of zero containing no other critical points, such that h(0) = 0 and (3.33)

'FIA S 0 < 'FIB\{o}

where A = h(IAo) and B = h(IBo) u {0}.

Proposition 3.34. If 'F has a cohomological local splitting near zero in dimension q, then Cq('F, 0) # 0. PROOF. (3.33) gives the commutative diagram (Do n U

'F6 n U\{0}

1

1

A

h(Ao)

hl

h1

IAo

A0

U\B

h

ki

k1

h(M\Bo)

IM\(IBo u {0})

h

M\Bo

h1

M\Bo

where i, j, j, k, k, 1, and the unlabeled maps are inclusions. Passing to the induced diagram on reduced cohomology in dimension q - 1, j* # 0 since j* # 0 by (3.32) and Proposition 2.14 (iv) and the h* are isomorphisms. Since the radial projection onto M is a homotopy equivalence, k*, and hence also k*, is an isomorphism. So the top middle square gives 1* # 0. On the other hand, the top left square gives l*i* = 0 since IAo, and hence also A, is contractible. So i* is not onto. Therefore 8, and hence also Cq('F, 0), is

3. CRITICAL POINT THEORY

62

nontrivial in the exact sequence n U\ {0})

Hq

b Hq('k°nU,V nU\{0}) of the pair ('F° n U,'F° n U\ {0}).

O

Remark 3.35. Definition 3.33 is a variant of the notion of homological local linking introduced by Perera [96], which also yields a nontrivial critical group at zero.

3.12. Even Functionals and Multiplicity In this section we assume that 'F is even and use the cohomological index to obtain multiple critical points. Let F denote the class of symmetric subsets of W\ {0}, and for k < din N, let

Fk={MeF:i(M)>, k}

(3.34)

and

Ck = inf sup 'F(u).

(3.35)

ME.Fk uEM

Since Fk Fk+1, Ck < ck+l, and since S6(0) e Fk for any 6 > 0 by (i8) and sup 'F(8(0)) --+ 0 as S \ 0 by continuity, Ck < 0.

Proposition 3.36. Assume that 4 is even. (i) If -co < ck = ... = ck+m,_I = c < 0 and 0 satisfies (C), then i(K`) >, m. In particular, if -co < Ck < .. < ck+,,,_1 < 0 and 0 satisfies (C), for c = ck,... , ck+m._1, then each c is a critical value and 'F has m distinct pairs of associated critical points. (ii) If -co < Ck < 0 for all sufficiently large k and 4D satisfies (C) for all c < 0, then Ck / 0. First a lemma.

Lemma 3.37. If c < 0 and 4D is even and satisfies (C), then there is an e > 0 such that c - 6 < Ck < - < ck+m-I < c + E i(K`) > m. (3.36) PROOF. Since K° is compact by (C), there is a d > 0 such that i(Ns(K`)) = i(KO) (3.37) by (i4), and there are an e > 0 and an odd map i e C(W, W) such that i7(V+E\N6(KC))

(3.38)

c 'FC-E

by Lemma 3.7. Then (3.39)

i('`+E)
i(BC) and hence intersects B by (i2), and > a on B, so Ck > a. Since A E= Fk+m_I and 4) C b on A, ck+m-1 < b. So the conclusion follows from Proposition 3.36.

Remark 3.39. Since A E Fk+m_I c Fk intersects B, inf 4)(B) < sup 4)(A).

3.13. Pseudo-Index The notion of a pseudo-index introduced by Benci [16] is useful for, among other things, obtaining multiple critical points of an even functional at positive levels.

Definition 3.40. Let F be as in the last section, M e F be closed, 0 C a < b < +co, and denote by r the group of odd homeomorphisms of W that are the identity outside D-1(a, b). Then the pseudo-index of M E F related to i, M, and r is defined by i*(M) = min i(ry(M) n M). 7Er

The following proposition lists some properties of the pseudo-index

i*:F--*Nu{0,oo}. Proposition 3.41. Let A, B e F.

3. CRITICAL POINT THEORY

64

(i) If A c B, then i*(A) < i*(B).

(ii) If rl e F, then i*(rl(A)) = i*(A). (iii) If A and B are closed, then

i*(A u B) < i*(A) + i(B). PROOF. (i) For each -y e r,

i(7(A) n M) < i(7(B) n M) by (i2). (ii) We have *(1!(A)) = min i(777(A) n M) = i*(A) 'yCr

since {7n : 7 E P} = F.

(iii) For each 7 e r, i(7(A u B) n M) = i((7(A) n M) u (7(B) n M))

6(7(A) n M) + i(7(B) n M) by (i5) and

i(7(B) n Al) < i(7(B)) = 2(B) by (i2).

For k < i(M) in N, let

Fk={MeF:i*(M)> k} and

ck = inf sup 0 and cF is even and satisfies (C), then there is an e > 0 such that c - E < C* -< " . < Ck+m-1 < c + e i(K`) >,m.

3.14. FUNCTIONALS ON FINSLER MANIFOLDS

65

PROOF. As in the proof of Lemma 3.37, there are b, e > 0 and t e IT such that (3.37) and (3.38) hold. So i*((I +e) < i*(4)c+e\No(Kc)) + i(N5(K`)) c i*(4)C-E) + i(K`)

by Proposition 3.41, and the conclusion follows as before.

Proposition 3.44. If M E F is bounded and radially homeomorphic to the unit sphere, U = IM; A, B E F with A c UC compact and B c M,

i(A)>k+m-1, for some k, m

i(M\B) Sk-1

1, and sup 4b(A) < a < inf 4)(B) C sup 4P(IA) < b,

and 4 is even and satisfies (C), for all c e (a, b), then a < ck 5 .. 5 ck+m-1 < b and hence there are m distinct pairs of critical points in l-1(a, b). PROOF. We apply Proposition 3.42. Each M e Fk satisfies

i(M n M) > i*(M) > i(M\B) and hence intersects B by (i2), and inf '(B) > a, so ck > a. If ry e IT, noting that -y1 A = idA since (b < a on A and applying Corollary 2.13 with Vi = ryIIA gives

i('y(IA)nM)>, i(A)> k+m-1. So i*(IA) > k + m - 1 and hence IA e Fti+m.-1 Since sup 4 )(IA) < b, then Ck+M-1 < b.

Remark 3.45. Since IA e Fk+ _1 c .F intersects B, inf'F(B) C sup 41, (1A).

3.14. Functionals on Finsler Manifolds A C1-manifold modeled on a Banach space V is a connected Hausdorff space M together with a collection {(U,, ep1\)}aeA, called an atlas on M, of pairs (UT, coa), called charts, such that (i) {Ua},EA is an open covering of M, (ii) for each A e A, Wa : U,, cpa(U,,) c V is a homeomorphism, (iii) for each A,li, e A, p, o coal (pa(U, n Uµ) cpµ(U, n Uµ) is a C'-diffeomorphism. We refer to Lang [64] for the basic constructions on M such as the tangent bundle TM and the cotangent bundle T*M. A Finsler manifold is a regular C'-Banach manifold M together with a continuous function TM --* [0, co), called a Finsler structure on TM, such that

(i) for each u e M, the restriction TTM at it is a norm,

IIu of

.

to the tangent space

3. CRITICAL POINT THEORY

66

(ii) for each u e M and C > 1, there is a trivializing neighborhood U of u such that

1111us11 -11.' 0, and 4' satisfies (PS), then there are c > 0 and rl e C([O,1] x M, M) satisfying (i) d(77(t, u), u) < bt V(t, u) e [0,1] x M,

(ii) 77(t, ) is the identity outside 4'CE0 ' for all t e [0, '1,

(iii) 4r(77(t, u)) < 4'(u)

V(t, U) E [0, 1] x M, (iv) 77(1, 4iV+E\Nj(K')) c P°-E.

Lemma 3.47 (Second Deformation Lemma). If -oo < a < b < +oo and 4' has no critical values in [a, b] and satisfies (PS), for all c e [a, b] n 1W, then 4'a is a deformation retract of 4'6.

Much of the theory of the previous sections can now be adapted to this setting.

Proposition 3.48. If c := inf 4'(M) is finite and 4' satisfies (PS),, then c is a critical value of 4'. PROOF. If not, let e > 0 and 77 e C([O, 1] x M, M) be given by Lemma 3.46 and take u e M with 4'(u) < c + E. Then 4(77(1, u)) < c - e by (iv), a contradiction.

Let A and B be disjoint nonempty subsets of M and

r = {y c C(CA, M) : 71 A = idA}. We say that A homotopically links B if

y(CA)nB#0 VyeP.

3.14. FUNCTIONALS ON FINSLER MANIFOLDS

67

Proposition 3.49. If M is a free 7G2-space and A and B are disjoint nonempty symmetric subsets of M such that

i(A) = i(M\B) < oo,

(3.41)

then A homotopically links B.

PROOF. If not, there is a -t e C(CA, M\B) with 71A = id A. Then

SA -. M\B, (u, t)

t E [0, 1]

ry(u, t),

7(-u, -t), t e [-1, 0)

is an odd map and hence

i(M\B) > i(SA) = i(A) + 1 by (i2) and (i6), contradicting (3.41).

Proposition 3.50. If A hornotopically links B, c:= inf

(3.42)

sup

4'(u)

ryEF uey(CA)

is finite, sup D(A) < inf'P(B), and 4D satisfies (PS)0, then c > inf'F(B) is a critical value of 4).

PROOF. If not, take eo < c - sup'P(A), let s and 11 be given by Lemma 3.46, and take rye P with sup'P(y(CA)) < c+s. Then 77(1, ) is the identity

on A by (ii), so 77(1, ) o y e F, but sup'P(r7(1, y(CA))) < c - s by (iv), contradicting (3.42). Now suppose that M is a free 7G2-space and 41, is even. The proof of the following symmetric deformation lemma can be found in Corvellec [32].

Lemma 3.51. If c e R, S > 0, and (D is even and satisfies (PS)0, then there are E > 0 and 77 e C([0,1] x M, M), with 77(t, ) odd for all t c [0, 1], satisfying (i), (iii), and (iv) of Lemma 3.46.

Let F denote the class of invariant subsets of M and for k < dim M in N, let Fk and ck be defined by (3.34) and (3.35), respectively. As in the proof of Lemma 3.37, it then follows from Lemma 3.51 that if c e R and 0 such that

i(K`)>, m. This in turn gives

Proposition 3.52. Assume that 9D is even. (i) If -CO < Ck = "' = Ck+m-1 = c < +co and lb satisfies (PS)c, then i(K`) in. In particular, if -co < ck S - - S ck+m_1 < +co and 'P satisfies (PS)0 for c = Ck, , Ck+m-1, then each c is a critical -

value and 4D has m distinct pairs of associated critical points.

(ii) If -oo < ck < +co for all sufficiently large k and $ satisfies (PS), then ck / +oo.

3. CRITICAL POINT THEORY

68

The following proposition gives the indices of the sublevel sets of 4'.

Proposition 3.53. Assume that 4' is even. (i) lick is finite and 4' satisfies (PS)0k, then i(M\cFCk) < k c i((D°k).

(3.43)

(ii) lick < ek+l are finite and 4' satisfies (PS)0 for e = ck, Ck+l, then

i(-`k) = i(M\4'a) = i(4'a) = i(M\4'ck+1) = k

Va E (Ck, Ck+1)

PROOF. (i) By Proposition 2.14 (iii), M\4'ek has a compact subset C e

F with i(C) = i(M\4'k). Then max4'(C) < ck since 4' is continuous, so C Tk and hence i(C) < k, and the first inequality in (3.43) follows. By (i4), 4'0k has a closed neighborhood N e F with i(N) = i((P°k). Since Kck is a compact subset of 4'Ck,

d := 2 dist(K°k, M\N) > 0. By Lemma 3.51, there are an'e > 0 and an odd map 71 E C(M, M) satisfying

d(rl(u),u) < b Vu e M,

E\Na(K°k)) C DCk-E

and hence

C DEk-E u N26(K`k) c N.

Taking an M e Tk with M c 4'`k+` then gives i(N) > i(4'`k+E) >, i(M) > k by (i2), so the second inequality in (3.43) also follows. (ii) By (3.43) and (i2),

k `k) S i(M\4'a) c i(4'a)

< k + 1.

Often M is of the form

M= In e W: 1(u) =1) with 1 a regular value of I E C' (W, ]k). Then M is a Cl-Finsler manifold by the implicit function theorem, and M is complete by the continuity of I. Moreover,

TuM = {v e W : (I'(u), v) = 0} = ker I'(u). Proposition 3.54. If 4' is a Cl-functional defined in a neighborhood of M and 4 its restriction to M, then the norm of 4f'(u) e TuM is given by (U) 11U = min 11 W' (u) - µ

(U)11

3.14. FUNCTIONALS ON FINSLER MANIFOLDS

69

PROOF. We have

RIME =

(q'(u),v)

sup

by (3.40)

veker I' (u) Iv1I=1

= min IIV(u) - p P(u) jAeR

by Lemma 3.55 below.

Lemma 3.55. If L, M E W*, then II LIkerM I` = min IIL - ttMII'. PROOF. For each p e II$,

II LlkerMII* = sup (L,v) S sup (L-pM,v) = IIL-hMII*. vekerM IvII=1

By the Hahn-Banach theorem, there is an L E W' such that L = L on ker M and

ILII'

- II LIkerM I'

Since ker (L - L) D ker M, L - Z = µM for some p e 1R, so II LIker M

IO'

= IIL - pMII `

.

CHAPTER 4

p-Linear Eigenvalue Problems In this chapter we study the p-linear eigenvalue problem (4.1)

Apu = ABpu

in W*, where A. satisfies (Al) - (A4), BV : W W* is (BI) (p - 1)-homogeneous and odd, (B2) strictly positive:

(By u, u) > 0

du # 0,

(B3) a compact potential operator, and A E R. We say that A is an eigenvalue of (4.1) if there is a u # 0 in W satisfying (4.1), called an eigenvector associated with A. Then an is also an eigenvector associated with A for any a # 0 by (A1) and (B1), and (Apu,u) (4.2)

A=

Bp u, u)

>0

by (A2) and (B2). The set o(Ap, Bp) of all eigenvalues is called the spectrum of the pair of operators (Ap , Bp ).

Example 4.1. In problem (1.9), the usual choice is

`Bpn V) = f

:

o

Op u = A Iulp-2 u in Q

U=0 i Then the first eigenvalue Al is positive, simple, and has an associated eigenon 852.

function Cpl that is positive in Il (see Lindqvist [68, 69]). Moreover, Al is iso-

lated in the spectrum o(-Ap), so the second eigenvalue A2 = inf o(-Ap) n (A1, oo) is well-defined. In the ODE case 71 = 1, where 1 is an interval, the spectrum consists of a sequence of simple eigenvalues Ak / oo, and the eigenfunction Wk associated with Ak has exactly k - 1 interior zeroes (see, e.g., Drabek [46]). In the semilinear PDE case n 2, p = 2 also, o(-a) consists of a sequence of eigenvalues Ak / oo. In the quasilinear PDE case re > 2, p :f 2, increasing and unbounded sequences of eigenvalues can be defined using various minimax schemes, but a complete list of the eigenvalues of -AP is still unavailable. 71

4. p.LINEAR EIGENVALUE PROBLEMS

72

Our setting also includes weighted eigenvalue problems as the next ex-

ample shows.

Example 4.2. In problem (1.9), taking (BP it, v) = JZ V(X)

jUlp-2 UV

with the weight V(x) > 0 a.e. and V e LS(Q) for some n

> -, PSn =1,

p>n

gives

-AP u = AV (x) lulp-2 a

u=0

in S2

onDQ.

This includes singular weights such as V(x) = lxl-9, 0 < q < min {p, n}.

4.1. Variational Setting Since (4.3)

(IP(u),u) = (Apu,u) =PIP(u),

zero is the only critical value of Ip and hence it follows from the implicit function theorem that

M := {u e W : II(u) = 1}

(4.4)

is a Cl-Finsler manifold. Moreover, M is complete, symmetric, and radially homeomorphic to the unit sphere since Ip is continuous, even, and p-homogeneous. By (1.6), (4.5)

l 1/p

inf

(Col

u s EM Ilull

ll IMP

(co/

By Proposition 1.2, the potential Jp of BP satisfying Jp(0) = 0 is given by

JP(u) = 1 (Bi,u,u) P

and is p-homogeneous and even. By (B2), Jp >, 0 and > 0 on W\ {0}. So the functional T(u) U C W\ {0}

4(u)'

is positive and its restriction

T=`PIm is CI. We will show that the critical values and the critical points of i are the eigenvalues and the eigenvectors of (4.1).

4.1. VARIATIONAL SETTING

73

Since

V(U)

_ _ )Z =

BP u,

.' (u) = Ap u,

it follows from Proposition 3.54 that the norm of V(u) e T,*M is given by III'(u)Ilu =mien

(4.7)

IIAAPU+'P(u)2BPUII*.

Lemma 4.3. Eigenvalues of (4.1) coincide with the critical values of ',

i.e., A is an eigenvalue if and only if there is a it e M such that W'(u) = 0 and W(u) = A.

PROOF. By (4.7), P(u) = 0 if and only if (4.8)

p e R. If (4.8) holds, applying it to u gives Y(u)2 (Ap

Y(u) c 0, - p(u)2 IP(u) -

u, u)

so (4.8) reduces to (4.1) with A = (u). Conversely, if A is an eigenvalue and it e M is an associated eigenvector, A = 1P(u) = JP(u)

(u)

by (4.2), so (4.1) implies (4.8) with p = -IY(u).

Example 4.4. In Example 4.1, M = 5 In E WD,P(Q) : 1 f I vulp ='I, w(u) _

P

st

IuiP,

P J11

Lemma 4.5.

JP(u) = 1

P st

l

lnIP

satisfies (PS).

PROOF. Let (uj) c M be a (PS), sequence, i.e., II'Y'(uj)11Ui '0.

'L(ug) -* c,

By (4.5), (uj) is bounded, so a renamed subsequence converges weakly to some u e W since W is reflexive, and BP uj converges to some L e W* for a further subsequence since BP is compact. Then (u.7) 'P

by Lemma 3.4, so c o 0.

P

= /BBuj,uj)

P

(L,u)

#0

4. p-LINEAR EIGENVALUE PROBLEMS

74

By (4.7),

µj Apuj +l (uj)2BPuj -.0

(4.9)

for some sequence (µi) c R. Applying (4.9) to uj gives µj + l (uj) 0, so µj -. -c # 0. Now applying (4.9) to uj - u gives (Ap uj, uj - u) - 0 since (BP uj, uj - u) 0 by Lemma 3.4, so uj -. it e M for a subsequence by (A4).

Lemma 4.5 implies that the set

Ea={uEM:I'(u)=0, lY(u)=A} of eigenvectors associated with A that lie on M is compact and that the spectrum o(Ap, BP) = {A e ]l8 : V(u) = 0, (u) = A for some u e M} is closed.

4.2. Minimax Eigenvalues We now construct an unbounded sequence of minimax eigenvalues for the eigenvalue problem (4.1). Although this can be done using the Krasnoselskii genus as usual, we prefer to use the cohomological index in order to obtain a nontrivial critical group and later construct linking sets. Let F denote the class of symmetric subsets of M, let

K= {keN:k(d}, and for k e K, let

Fk={MEF:i(M)>, k} and Ak = inf sup 11(u). MEFk uEM

Since Fk D Fk+1, Ak < Ak+l, and since the intersection of M with any k-dimensional subspace of W is a compact set in Fk by (is), Ak is finite. When d < oo we set Ad+1 =

OC)

for convenience.

Theorem 4.6. Assume (A1) - (A4) and (B1) - (B3). Then (Ak)kEK is a nondecreasing sequence of eigenvalues of (4.1).

(i) If Ak = "' = Ak+m-1 = A, then i(Ex) >, in. In particular, there are d distinct pairs of eigenvectors on M. (ii) The smallest eigenvalue, called the first eigenvalue, is IP(u) > 0. u#0 ,IJ(u)

Al = min qf(u) = min ueM

4.2. MINIMAX EIGENVALUES

(iii) We have i(.M\ (Yak) < k

75

i(1PAk). If AAk < A < Ak+1, then

i(M\t,) = 2(P) = 2(M\'Yak+1) = k. (iv) If d = co, then Ak / oo. PROOF. (i), (iv) Follow from Proposition 3.52. (ii) The first equality holds since T1 contains all antipodal pairs of points in M by Proposition 2.14 (i), and the second follows from homogeneity. (iii) Follows from Proposition 3.53.

The spectrum o(Ap, Bp) may possibly contain points other than those of the sequence (Ak), and the eigenvalues Ak may possibly be different from

the standard ones

ak =

inf

sup ql(u)

MET uEM 7±(M)->k

where

'Y+ (M) := sup {k > 1 : 3 an odd continuous map Sk-1

M}

and

-y-(M) := inf {k >, 1: 3 an odd continuous map M

Sk-1}

are the genus and the cogenus of M e .F, respectively.

Proposition 4.7. We have (z) 4i = A1,

(ii) Ak SAk2. PROOF. (i) Since both the genus and the cogenus of an antipodal pair of points in M is 1, al = min lF(u) = Al uEM

by Theorem 4.6 (ii). (ii) If M e T and there are odd continuous maps then

Sk+-1

-> M

Sk -1

k+ I/1,(u) - A Jp(u) - C

I

1 - 1)

-. oo as

jju1jP - C

by (5.4)

by Theorem 4.6 (ii) and (1.6)

- Co. Every (PS) sequence is bounded by coercivity and hence has a convergent hull

subsequence by Lemma 3.3. (ii) 1i > -C by (5.5), and then the rest follows from Corollary 3.19.

Example 5.2. In Example 4.4, (5.4) holds if

F(x, t) 5 A ltjP + C V(x, t) P

for some A < Al and C > 0.

5.2. Asymptotically p-Linear Case We strengthen (5.2) by assuming that

f = ABp - g

(5.6)

for some A > 0 and a compact potential operator g : W W* satisfying g(u) = o(llullp 1) as llull - oo.

(5.7)

Then F(u) = 'DA(u) + G(u) where Fa is given by (4.10) and (5.8)

G = A Jp - F

(5.9)

is the potential of g satisfying G(0) = 0. By Proposition 1.2 and (5.7), (5.10)

C(u) = I (g(tu), u) at = o(IIull') as dull -> oo. I

Example 5.3. In problem (1.9), assume that

f(x,t) = AItIP 't -g(x,t) for some g e Car(Q x R) satisfying

g(x, t) = o(ltl"-1) as tl

co; uniformly in x.

Then we have

-Ap u = A lulp-2 u - g(x, u)

{

u=0

in S2

on aU,

5.2. ASYMPTOTICALLY p-LINEAR CASE

81

and (5.6) and (5.7) hold with (g(u), v) =

f2 g(x, u) v,

G(u) =

f

G(x, u)

where

G(x,t) = f g(x, s) ds. 0

To verify the (PS) (resp. (C)) condition for D, it suffices to show that every (PS) (resp. (C)) sequence is bounded by Lemma 3.3, and the following lemma is useful for this purpose.

0 and pj :_

Lemma 5.4. If D'(uj)

Ik

II

-* co, then a subsequence of

uj := 'uj/pj converges to an eigenvector v associated with A.

PROOF. We have 11 ujll = 1 for all j, so a renamed subsequence converges

weakly to some v, since W is reflexive, and Bpvj converges in W* for a further subsequence since Bp is compact. Then

uj - v) _

+(4"(uj)-g(ui),wj-u)

A/Bpuj wj

y0

by Lemma 3.4 and (5.7), so vj -> u for a subsequence by (A4). Now passing

to the limit in Ap uj = A Bp uj +

V (uj) -Ig (uj ) Yj

gives Ap u = A Bp u, and u # 0 since Hull = 1.

Since (5.9) and (5.10) imply that (5.4) holds with a slightly larger A, here we assume that Ak < A < Ak+l for some k e ftC such thatAk < Ak+1. Let M and "P be as in Section 4.1 and let AO = yak and Bo = Wak+i Then

i(Ao) = i(M\Bo) = k by Theorem 4.6 (iii), so A = RAo cohomologically links B = 7r7 (Bo) u {0} in dimension k - 1 for any R > 0 by Proposition 3.26. By (5.8), (4.14), and (1.6),

1)

4'(Ru) < G(Ru) -

(5.11)

and

/

4'(u) > I 1 -

(5.12)

k+1 Lemma 5.5. If $ satisfies (C),

RP,

) Co Iullp + G(u), P

u e AO

u e B.

1

(5.13)

Rimes

- 1) RP] \k

f G(Ru) - 1 L

= -co, uniformly in u e AO,

5. EXISTENCE THEORY

82

and (5.14)

inf

r/

l

c0 IluIIp

1

I

k+1

L

P

+ G(u)J > -co,

then C°(4i,oo) = 0, Ck(,D,co) # 0, and (1.1) has a solution u with Ck(-P,u) # 0.

PROOF. -b is bounded from below on B by (5.12) and (5.14), and for any a < inf 'F(B) and sufficiently large R, (3.31) holds by (5.11) and (5.13), so the conclusion follows from Proposition 3.25.

Lemma 5.6. If A > Ak, then (5.13) holds, and if A < Ak+i, then (5.14) holds.

PROOF. The first assertion follows since llimo

G(RRu)

= 0, uniformly in it e A0

by (5.10) and (4.5), and the second follows similarly.

Nonresonance. We say that (1.1) is nonresonant if A 0 o-(Ap , By) in (5.6).

Theorem 5.7. Assume (A1) - (A4), (Bi) - (B3), and (5.6) and (5.7) with A E (Ak, Ak+l)\Q(Ap, Bp) Then -

(i) P satisfies (PS),

(ii) C°(p, cc) = 0,

oo) # 0, and (1.1) has a solution it with

0.

PROOF. (i) Since A 0 v(Ap, Bp ), every (PS) sequence is bounded by Lemma 5.4 and hence has a convergent subsequence by Lemma 3.3. (ii) Follows from Lemmas 5.6 and 5.5.

Resonance. We say that (1.1) is resonant if A e o,(Ap , Bp). Then we make an additional assumption on the non-p-homogeneous part of cF given by 1 (4i'(u), u) = G(u) - 1 (9(u), u) H(u) _ -P(u) - P

in order to ensure that D satisfies the (C) condition. Note that (H(uj)) is bounded for every (C) sequence (uj). Denoting by.NV the class of sequences

(uj) c W such that pj :_ INujIi -> oo and uj := uj/pj converges weakly to some u # 0, we assume one of (H±) H is bounded from below (resp. above) and every sequence (uj) e N has a subsequence such that

H(tuj)

±co Vt '> 1.

In particular, no (C) sequence can belong to N.

5.2. ASYMPTOTICALLY p-LINEAR CASE

83

Lemma 5.8. If (H±) holds, then G is bounded from below (resp. above) and every sequence (uj) e N has a subsequence such that ±ao.

G(uy)

(5.15)

PROOF. We have

p H(tu)

G(tu)

d

dt

=

tP

lim t-.m

tP+1

G(tu) = 0 tp

pH(tu) G(u) = J 1

dt.

f+1 dt = 1, inf H 5 G C sup H and (5.15) for the subsequence in (H+) follows from Fatou's lemma. We can now prove

Theorem 5.9. Assume (A1) - (A4), (BI) - (B3), and (5.6) and (5.7). Then (i) 4) satisfies (C) if (H+) or (H_) holds, (ii) D is bounded from below, C"(4), oo) z 6qo Z2, and 4) has a global minimizer u with C'](4), u)

6go7G2 if A = Al and (H+) holds,

(iii) Co(4), co) = 0, C'(4), co) # 0, and (1.1) has a solution u with Ck(4), u) # 0 in the following cases: (a) A E [Ak, Ak+1) and (H_) holds, (b) A E (Ak, Ak+1] and (H+) holds.

PROOF. (i) If a (C) sequence is unbounded, then Lemma 5.4 gives a subsequence that belongs to N, contradicting (H+). (ii) 4i = 4a, + G is bounded from below since 4ia, = I, - Al J, >, 0 by Theorem 4.6 (ii) and G is bounded from below by Lemma 5.8, and then the rest follows from Corollary 3.19. (iii) We apply Lemma 5.5. (a) Since A < Ak+1, (5.14) holds by Lemma (5.6), and since A >, Ak, it suffices to show that

lim G(Ru) = -co, uniformly in u e A0

R,.

oo such to verify (5.13). If not, there are sequences (u.,) c Ao and Rj that (G(Rj uj)) is bounded from below. Then pj := R7 11u111 -. oo by (4.5), a renamed subsequence of v,j := R. ujlpj = uj/ IIu, 11 converges weakly to some

5. EXISTENCE THEORY

84

u since W is reflexive, and BA converges to some L e W*.for a further subsequence since BP is compact. We have

(L, u) = lim BPVj, uj)

= lim .

by Lemma 3.4

P

'p(ui) 1v9 IIP

by (4.5)

by Theorem 4.6 (ii)

and hence u # 0, so (R uj) e N. But, then G(Rj uj) -* -oo for a subsequence by Lemma 5.8, a contradiction. (b) Since A > Ak, (5.13) holds by Lemma (5.6), and since A 5 Ak+I and G is bounded from below by Lemma 5.8, (5.14) holds as well.

Example 5.10. In Example 5.3,

H(u) =

Jfn

H(x,u)

where

H(x, t) = G(x, t) -

g(x, t) t

is the non-p-homogeneous part of G(x, t). pWe claim that (H±) holds if (5.16)

H(x, t) 3 (resp. v a.e. and hence H(tuj) _> (resp. W\ {0}

,

(1 - t) u + tTa(irs(u)) irs(u), (u, t))

in,

u e (W\

u e V.

We can now prove

Theorem 5.13. Assume (Al) - (A4), F is bounded from below and satisfies (5.3), and H is bounded from above for some p > p. Then (i) 4) satisfies (PS), (ii) Cq(4),00) ^ bgd Z2, (iii) 4) is bounded from above and has a global maximizeru with Cq(4), u) bgdZ2 if d < oo.

5. EXISTENCE THEORY

86

PROOF. (i) If (uj) is a (PS) sequence, then (1.6) gives 1

co

P

1

/ 1

iiu7 lip

E) Ip(u7)

(I

_ 4)(ui) -

- (4)'(u9),ui) + HM(ui)

c oGiuj 11) + 0(1),

so (uj) is bounded and hence has a convergent subsequence by Lemma 3.3. (ii) Since any a < ao is less than all critical values and 4) is unbounded from below by (5.18), Cq(4p,00) x HQ-1($a)

by Proposition 3.15 (ii)

z Hq-r(S)

by Lemma 5.12

8qd Z2

by Example 2.5.

(iii) Since S is compact, so is a = {tu : u e S, 0 < t < Ta(u)} and hence 4) is bounded there. Since 4 < a outside (ba, then 4) is bounded from above everywhere, and the rest follows from Propositions 3.17 and 3.18.

Example 5.14. In problem (1.9), we claim that all the hypotheses of Theorem 5.13 are satisfied if 0 < F(x, t) 5

(5.19)

1

lti >,T

f (x, t) t,

W

for some p > p and T > 0. By (1.10), f (x, t), and hence also F(x, t), is bounded on bounded t intervals. Integrating (5.19) gives

F(x, t) > c(x) Its" - C V(x, t) where c(x) = min F(x,±T)/Tµ > 0 and C > 0, so F((tu)

J

t"-p

f c(x) gulp -

C 1",

o0 as t - 00 Vu # 0

and F(u) >, -C 101 where JQJ is the volume of Q. Finally, since HM,(x,t)

F(x,t) - µ f(x,t)t

is S 0 for tj > T by (5.19) and bounded for Itl < T, HM(u) = L HM(x, u)

is bounded from above.

Remarks 5.15. In the semilinear case p = 2 of Example 5.14, condition (5.19) was introduced by Ambrosetti and Rabinowitz [7] and the critical groups at infinity were determined by Wang [132].

CHAPTER 6

Monotonicity and Uniqueness In this short chapter we give simple sufficient conditions for equation (1.1) to have at most one solution and show how they can be verified in applications.

Definition 6.1. An operator T : W

W* is monotone (resp. strictly

monotone) if

(T(u) - T(v), u - v) > (resp. >) 0 Vu # v. Theorem 6.2. If AP is strictly monotone and -f is monotone, then (1.1) has at most one solution.

PROOF. If both ul and u2 are solutions, then (Ap ui - Ar, u2, ui - 112) = (f (ui) - f (n'2), u1 - u2) ( 0 and hence ul = 162.

A.

The following lemma is useful for verifying the strict monotonicity of

Lemma 6.3. If (Apu,v) 5

IIvII

IIuIV'

Vu,v E W

and the equality holds if and only if au = /3v for some a, )3 >, 0, not both zero, then Ap is strictly monotone. PROOF. As in (1.8),

(Apu-Apv,u-v) >- (IIuIIp-1-IIvrrr-')(IIuII

- IIvII)

0.

If the leftmost term is zero, equality holds throughout and hence (Ap u, v) _ 1

IIuIIp

11v 11, (Apv,u) = Ilvllp

1

lull, and IIuII = 11v 11. Then an = /3v for some

a, 0 > 0, not both zero, so either a = v = 0, or a = /3 > 0 and hence

u=v#0.

Example 6.4. In problem (1.9), -f is clearly monotone if f (x, t) is nonincreasing in t, and now we show that the strict monotonicity of Ap follows from Lemma 6.3. By Schwarz and Holder inequalities,

(Apu,v) = f IVulp-2 Vu _ Vv 5 87

f IVul" lVvl - IIuIIp-1 IIvII

88

6. MONOTONICITY AND UNIQUENESS

Clearly, equality holds throughout if au = Qv for some a, (3 , 0, not both zero. Conversely, if (Ap u, v) = IIuiIP-1 iIvjj, equality holds in both inequalities. The equality in the Holder inequality gives

a IVul = Q IVvI a.e.

for some a, /3 ? 0, not both zero, and then the equality in the Schwarz inequality gives

a Du = /3 Vv a.e.,

so an = /iv.

CHAPTER 7

Nontrivial Solutions and Multiplicity In many applications f(0) = 0 and hence equation (1.1) has the trivial solution u = 0, and we are interested in finding others. Throughout this chapter we assume that this is the case and there is only a finite number of solutions.

7.1. Mountain Pass Solutions First we obtain nontrivial mountain pass solutions of (1.1) in the asymptotically p-linear and p-superlinear cases assuming that either F(u) < al Jp(u)

(7.1)

Vu E Bp(0)

for some p > 0, or

F(u) for some 05A, Al for all u # 0 by Theorem 4.6 (ii), if (7.1) holds, then 4) (u) >_ Ip(u) - Al Jp(u) > 0 = -D (0)

Vu E Bp (0),

and if (7.2) holds, then

II(u)

>,1 -

Al

+ o(1) as u -. 0

by (1.6). Proposition 3.18 gives the critical groups.

Theorem 7.2. Assume (Al) - (A4), (Bi) - (B3), (5.6) and (5.7), and (7.1) or (7.2). Then (1.1) has a mountain pass solution u # 0 with 415(u) > 0 in the following cases: (a) A c (Ak, Ak+l)\Q(Ap, Bp), (b) A E [Ak, Ak+l) and (H_) holds, (c) A E (Ak, Ak+l] and (H+) holds.

PROOF. Follows from Corollary 3.31 since C°(F, 0) x 7L2 by Lemma 7.1 and C°(,D, oo) = 0 and 4 satisfies (C) by Theorem 5.7 in case (a) and Theorem 5.9 in cases (b) and (c). 89

7. NONTRIVIAL SOLUTIONS AND MULTIPLICITY

90

Theorem 7.3. Assume (Ai) - (A4), (B1) - (B3), F is bounded from below and satisfies (5.3) and (7.1) or (7.2), and Hp, is bounded from above for some

p > p. Then (1.1) has a mountain pass solution ul # 0 with 4?(ul) > 0. If 2 < d < oo. then there is a second solution u2 # 0 that is a global maximizer of 4? with 4?(u2) > 0.

PROOF. Since Co((1,0) .: Z2 by Lemma 7.1 and C°(4t,co) = 0 and 4? satisfies (PS) by Theorem 5.13, (b has a mountain pass point ul # 0 with 4?(uI) > 0 by Corollary 3.31. When 2 5 d < co, there is a global maximizer u2 with CQ('F, u2) x Sgd7G2 by Theorem 5.13 (iii), 'D (U2) >, 4?(ui) > 0, and O U2 # uI since C1(4?,u2) # Cl(4),uI).

Example 7.4. When p > n in Example 4.4, (7.1) holds if

F(x,t) < Pl Itip,

(7.3)

ItI 5 6

for some 6 > 0 by the Sobolev imbedding Wo''(S2) - C(Si). When p < n, we strengthen (7.3) to

F(x4) c A ItID,

ti C 6

p

for some 0 < a < A1. Then

F(x,t) p. Then (1.1) has a solution ul # 0 with either

(D(ul) < 0 and Ck-1(4',ul) # 0, or 4'(ul) > 0 and Ck+1(4 u.l) # 0. If k + 2 < d < oo, then there is a second solution u2 # 0 that is a global maximizer of 4' with 4(u2) > 0.

PROOF. Since Ck(4', 0) # 0 by Lemma 7.5 and Ck(4', CC) = 0 and 4' satisfies (PS) by Theorem 5.13, 4' has a critical point ul # 0 with either

,D(ul) < 0 and C'-1(4',ul) # 0, or 4'(u1) > 0 and Ck+I(4',ul) # 0 by Proposition 3.28 (ii). When k + 2 < d < oo, there is a global maximizer u2 with Cq(4',u2) bgd72 by Theorem 5.13 (iii), 4'(u2) > 4'(0) = 0 since Ck(4', 0) # 0 and hence the origin is not a global maximizer by Proposition 3.18, and u2 # u1 since Ck-1(4 U2) = Ck+1((pu2) = 0

Example 7.8. When p > n in Example 4.4, (7.4) holds if pk ItIP
0. PROOF. By (7.6), .D(tu) I 1 -

A +o(1)) rP as r Ak+1

Since

i(Ao) = i(M\Bo) = k

0, u e Bo.

7.4. ODD NONLINEARITIES

93

by Theorem 4.6 (iii), A homotopically links B when R > r by Proposition 3.23, and since A $ o-(Ap, Br ), 4? satisfies (PS) by Theorem 5.7. Applying Proposition 3.21, c >, inf 4?(B) > 0 defined by (3.27) with g : A c W and X = IA is a critical value of 4?, and there is a corresponding critical point

onBwhenc=0. Example 7.10. In Example 4.4, (7.6) holds if F(x, t) >'

Ak

p

Itlp

d(x, t).

Remarks 7.11. Theorem 7.9 in the p-Laplacian case is due to Perera and Szulkin [105]. The semilinear case p = 2 is a well-known result of Amann and Zehnder [4].

7.4. Odd Nonlinearities Finally we assume that f is odd, so that 4? is even, and obtain multiple solutions of (1.1) using Propositions 3.38 and 3.44.

Theorem 7.12. Assume (Al) - (A4), (Bi) - (B3), (5.4), and f is odd. If F(u) 3 A Jp(u) + o(IIulI1) as u -+ 0 for some A > Am, then (1.1) has m distinct pairs of solutions at negative (7.9)

levels.

PROOF. We apply Proposition 3.38, taking A = r P- with r > 0 sufficiently small and B = W. We have i(A) >, m by Theorem 4.6 (iii) and i(B") = 0 by (i1). By (7.9) and (4.5),

4i(ru)5-C

-1+0(1))VP

as r-->0,ue

so sup 4?(A) < 0 when r is sufficiently small. By Theorem 5.1, 41, is bounded

from below, and hence inf 4i(B) > -co, and satisfies (PS). The conclusion follows.

When (1.1) is asymptotically p-linear we only consider the nonresonant case for the sake of simplicity.

Theorem 7.13. Assume (AI) - (A4), (Bi) - (B3), (5.6) and (5.7) with A 0 a(Ap Bp), and f is odd. ,

(i) If A < Ak and (7.10) F(u) ? A Jp(u) + o(jIuIIP) as u 0 for some A > Ak+.,,,_l, then (1.1) has m distinct pairs of solutions at negative levels. (ii) If A > Ak+m-I and (7.11) F(u) < AJp(u) + o(IIuIIp) as it - 0

for some A < Ak, then (1.1) has m distinct pairs of solutions at positive levels.

7. NONTRIVIAL SOLUTIONS AND MULTIPLICITY

94

PROOF. Since A 0 a(AP, Br ), 4 satisfies (PS) by Theorem 5.7.

(i) We apply Proposition 3.38, taking A = r l Ik+n.-, with r > 0 sufficiently small and B = 7r j (IWTk) u {0}. We have i(A) > k + m - 1 and

i(BC) = i(1rc (M\tAk)) = i(M\Pak) < k - 1 by Theorem 4.6 (iii). By (7.10) and (4.5),

a

4,(ru)' L

Tk) p + 0(1)] IIuIIP

\1

as dull -. cc, u e B,

so inf 4)(B) > -oo. (ii) By Proposition 2.14 (iii), M\WA has a compact subset C cF with

i(C) =

We apply Proposition 3.44, taking A = RC and B = r 41ak c rM with r > 0 sufficiently small and R > r sufficiently large. We have

i(A) = i(C) = k+m-1 and i(rM\B) 5 k - 1 by Theorem 4.6 (iii). By (5.8) and (4.14),

CRu)

A

maxW(C)

-1-

G(R u)

)

RP,

u e C, R > 0,

and by (5.10) and (4.5),

G(Ru) Rv

-> 0 as R , cc, uniformly in u e C.

Since max (C) < A by the continuity of F-, it follows that sup 4 (A) < 0 when R is sufficiently large. By (7.11) and (4.5), iD(ru)

(1 - a +0(1) ) r7' as r - 0, u e Yak, k

so inf 4D(B) > 0 when r is sufficiently small.

0

Theorem 7.14. Assume (A1) - (A4), (Bi) - (B3), F is bounded from below and satisfies (5.3), HH is bounded from above for some p > p, and f is odd.

If F(u) < a JP(u) + oU uIl") as u -. 0

for some A < Ak, then (1.1) has d - k + 1 distinct pairs of solutions at positive levels. If d = w, then there is an unbounded sequence of positive critical levels.

7.4. ODD NONLINEARITIES

95

PROOF. We will show that $ has in distinct pairs of critical points in $-1(0, oo) for any m S d - k + 1 in N. We apply Proposition 3.44, noting that 4D satisfies (PS) by Theorem 5.13, and taking A to be the intersection of SR(O) with any (k+m-1)-dimensional subspace of W and B = r Yak c rM,

with r > 0 sufficiently small and R > (p/co)1/1'r sufficiently large. A is a compact symmetric set with index k + m - 1 by (i8) and i(rM\B) < k - 1 by Theorem 4.6 (iii). By (5.18), sup 4 (A) S 0 when R is sufficiently large, and as in the proof of Theorem 7.13 (ii), inf b(B) > 0 when r is sufficiently small. When d = oo, the sequence of critical levels in (0, oo) given by Proposition

3.44 is unbounded by Proposition 3.42 (ii).

Remark 7.15. Theorem 7.13 in the p-Laplacian case is due to Perera and Szulkin 11051.

CHAPTER 8

Jumping Nonlinearities and the Dancer-Fucik Spectrum Consider the problem

Apu=aB. u+bBB u in W*, where Ap satisfies (Al) - (A4), BBi : W W* are (Bl) (p - 1)-homogeneous, (8.1)

(BZ) nonnegative:

(BPu,u)>0 VUEW, (B3) compact potential operators,

(B4) Bp + Bp = Bp, and a, b e R. The set E(Ap , Bp , Bp) of all points (a, b) e R2 such that there is a it # 0 in W satisfying (8.1) is called the Dancer-Full spectrum of the triple of operators (Ap, Bp , Bp ). By Proposition 1.2, the potentials Jp of BP satisfying Jp (0) = 0 are given by

JP(u)=1(BFu,u) P

and are p-homogeneous. By (B2 ), Jp (u) > 0 for all u E W. Moreover, (8.2)

JP (u) + Jp (u) = JJ(u) > 0 Flu # 0

by (B4) and (B2), so either Jp (u) > 0 or Jy (u) > 0 when it # 0.

If

(a, b) e E(Ap , Bp , By) and it # 0 satisfies (8.1), applying it to it and using (8.2) gives

Ip(u) =a J4 (u) + b Jp (u) < max {a, b} Jp(u) and hence max {a, b} > Al by Theorem 4.6 (ii), so E(Ap, Bp , By) c {(a, b) C R2 : max {a, b} > Ai }.

By (B4), E(Ap, Bp , Bp) contains the set { (.F, A)

:

particular, the sequence of points ((Ak,Ak))kEK'

Example 8.1. In Example 4.1, the usual choices are (Bp U, V) =+Jn (u±)p-1v 97

A E a(Ap, Bp) }, in

8. JUMPING NONLINEARITIES AND THE DANCER-FUCIK SPECTRUM

98

where u± = max {± u, 0} are the positive and negative parts of u, respectively:

AP u = a (u+)p-I - b (u

)P-I

mQ

u=0 i The Dancer-Fucfk spectrum EI, was introduced in the semilinear case p = on On.

2 of this problem by Dancer [38, 39] and Fu, 1ik [51], who recognized its significance for the solvability of (1.9) when

f (x, t) = at'- - bt- + o(jtI) as

Itl

oo, uniformly in x.

In the ODE case n. = 1, Fhcffc showed that E2 consists of a sequence of hyperbolic like curves passing through the points (Ak, Ak), with one or two curves going through each point, and Drabek [46] has recently shown that Ep has this same general shape for all p in this case. In the PDE case u > 2, E2 consists locally of curves emanating from the points (Ak, Ak), in particular, contains two strictly decreasing curves, which may coincide, such that the points in the square (Ak_I, Ak+I) x (Ak-i, Ak+l) that are either below

the lower curve or above the upper curve are not in E2, while the points between them may or may not belong to E2 when they do not coincide (see Cac [22], Cuesta and Gossez [37], de Figueiredo and Gossez [41], Margulies and Margulies [77], and Schechter [119]). Since c°I solves

Op u = aI (u+)P-I - b (u-)P-I in 52 {

l

u=0

on3Q

for arbitrary b and -cpI solves Op u = a (u+)P

t

I

- aI (u )P-I

u=0

in 52

onaQ

for arbitrary a, Ep contains the two lines aI x R and R x AI. A first nontrivial curve in Ep passing through (A2, A2) and asymptotic to aI x 1R and 1R x aI at infinity was constructed using a mountain pass scheme by Cuesta, de Figueiredo, and Gossez [36]. More recently, unbounded sequences of decreasing curves of Ep, analogous to the lower and upper curves of Schechter in the semilinear case, have been constructed using various nlinimax schemes

by Cuesta [35], Micheletti and Pistoia [83], and Perera [101]. We close this introductory section with

Proposition 8.2. The spectrum E(Ap, By , BP) is closed. PROOF. Let the sequence ((aj, bj)) c E(Ap, B,+, By) converge to (a, b) e lR2 and let uj # 0 satisfy (8.3)

Ap uj = aj BP uj + bj B-uj.

8.1. VARIATIONAL SETTING

99

Using (Al) and (Bl) to replace uj with uj/ Iuj 11 if necessary, we may assume that jjujIj = 1, so a renamed subsequence converges weakly to some u since W is reflexive, and By uj converge in W* for further subsequences since BP are compact. Then (Ap uj, uj

- u) = aj (BP uj, uj - u) + bj (BD uj, uj - u) - 0

by Lemma 3.4, so uj

u for a subsequence by (A4). Now passing to the

limit in (8.3) shows that u satisfies (8.1), and u # 0 since hull = 1, so (a,b)eE(Ap,BP ,Bp ).

8.1. Variational Setting For each s > 0,

By := BP + s By is a (p-1)-homogeneous compact potential operator from W to W* by (Bl ) and (B3 ), and the p-homogeneous potential JP of BP satisfying 0 is given by Jp (u) =

P

(BP u, u) = JP (u) + s Jp (u).

Since either Jp (u) > 0 or Jp (u) > 0 when u # 0, JD(u) > 0 for all u # 0,

so the functional

T-(u) = Jp(u)' is positive and its restriction

u E W\{0}

to the manifold Nl defined by (4.4) is C1. We will show that the points of E(Ap , Bp , By) that are on the ray b = so, a >, 0 are of the form (c, se) with c a critical value of 1y. Since

V8(u) = -`FS(u)z BP u,

it follows from Proposition 3.54 that the norm of 'F (u) e (8.4)

is given by

l-s(u)Ilu = mm en

Lemma 8.3. The point (c, sc) e E(Ap, Bp , Bp) if and only if c is a critical value of '3. PROOF. By (8.4), YS(u) = 0 if and only if

µApu+ii,(u)2Byu = 0

(8.5)

for some p e R. If (8.5) holds, applying it to it gives s

a=

u

-W3(u)2 (BP u,u

3

'3(u)2 IPu) = -W3(u) < 0,

S. JUMPING NONLINEARITIES AND THE DANCER-FUCIK SPECTRUM

100

so (8.5) reduces to

Apu = cBy u

(8.6)

where c = ;P-, (u). Conversely, if it e M solves (8.6),

c=

IP(u) p(U)

so (8.6) implies (8.5) with p =

Example 8.4. In Example 8.1, J (u) = P

f P

(u+)p

`I' s (u)

f

P

= r

(u+)p + s (u-)p

Lemma 8.5. 's satisfies (PS). PROOF. Let (uj) c M be a (PS), sequence. By (4.5), (uj) is bounded, so a renamed subsequence converges weakly to some it e W since W is reflexive, and BP uj converges to some L e W* for a further subsequence since BP is compact. Then ' (

1P s u

)=

P

P

Bnu UJ

(L u)

#0

by Lemma 3.4, so c # 0. By (8.4), pj Ap uj + &(uj)2 By uj

(8.7)

0

for some sequence (pj) c R. Applying (8.7) to uj gives PPj + Ps(uj) - 0, so µj -. -c # 0. Now applying (8.7) to uj - it gives (Ap uj, uj - u) --* 0 since (By uj, uj u) 0 by Lemma 3.4, so uj -. it e M for a subsequence by

-

(A4).

8.2. A Family of Curves in the Spectrum We now construct an unbounded sequence of decreasing (continuous) curves in E(Ap , Bp , BP ).

Lemma 8.6. If sl < 82, then xpsi ! q/s21

sl `Ysl < s2 ` '32.

PROOF. We have

`I'si-`I's2=(S2-Sl)JP'sikPs230 and

81 T,, - 82 T12 = (S1 - 82) JJ Wsi'I`s2 5 0.

8.2- A FAMILY OF CURVES IN THE SPECTRUM

101

First we construct a curve Cl passing through the point (Al, Al). Let

c1(s) = EnAf s(u) and

C1 = {(cl(s),scl(s)) : s > 0}.

Theorem 8.7. Assume (Al) - (A4), (B1) - (B3), and (Bl) - (B4 ). Then Cl is a decreasing curve in E(Ap, Bp +, BP) passing through (Al, A1).

PROOF. cl(s) is a critical value of Ws by Proposition 3.48 and hence C1 c E(Ap, Bp +, By) by Lemma 8.3. If 81 C s2, then (8.8)

Ci(Sl) ! C1(82),

Si C1(81) 1< S2 C1(s2)

by Lemma 8.6, so Cl is decreasing. To see that it is continuous, note that (8.8) implies 82

C1(s2) < C1(s1) (

C1(82), 81

Sl C1(s1) < C1(82) < Cl(S1) 82

and hence c1(sl) -* cl(s2) as 81 / S2 and cl(s2) -* c1(s1) as 82 N sl. Since

Jp +JD = Jp by (8.2), `' is the functional >Y defined by (4.6), so cl(1) = Al by Theorem 4.6 (ii) and hence C1 passes through (A1, A1).

Now we construct a curve Ck in the spectrum for each k 3 2 in IC such that Ak > Ak-1. Let Ak = tp ak-i rk = {ry E C(CAk, M) : 7I Ak = id Ak}+

Ck(s) = inf

sup

'P,9(u),

-YEFk uE7(CAk)

and Ck = {(Ck(S), s Ck(S)): Ak-1/Ak < S < Ak/Ak-1}.

Theorem 8.8. Assume (Al) - (A4), (Bi) - (B3), and (B1) - (B4 ). Then Ck is a decreasing curve in E(Ap, BP , BP) and ck(1) 3 Ak. PROOF. Let Bk = 'Yak. Then

i(Ak) = i(M\Bk) = k - 1 by Theorem 4.6 (iii) and hence Ak homotopically links Bk by Proposition 3.49. Since

max is, 1}

min {.s, 1

when Ak_i/Ak < s < Ak/Ak-1,

sup I3(Ak) 5

Ak

mAk

< < inf q1s(Bk) in {s,l 1} max {s, 1}

It follows from Proposition 3.50 that Ck(S) >,

Ak

max{s,1}

8. JUMPING NONLINEARITIES AND THE DANCER-FUNK SPECTRUM

102

is a critical value of 'Ifs and hence Ck c E(AP, Bp ,BP ). By Lemma 8.6, S1 Ck(SI) 5 82 Ck(s2)

ek(81) % ek(s2),

Vs1 0 such that Pa'b ))1

(JPa'b))1

V(a', b') e B5(a, b).

Since the set R2\E(Ap, Bp , BP) is open by Proposition 8.2, for all sufficiently small 8 > 0, B5(a, b) c R2\E(Ap , BP , Bp) and hence 1 is a regular value of Pa'b) for all (a', b') a B5(a, b) by Lemma 8.10. Lemma 8.11 gives the following stronger version.

Lemma 8.14. If (a, b) e R2\E(Ap, Bp , Bp ), then there is a 6 > 0 such that JPab) has no critical values in [1, 1 + 36/a1] for all (a', b') e B5 (a, b).

PROOF. If not, there are sequences 6l \ 0, (o'l,bj') e Bo,(a,b), and (u1) c M such that (ab)(ul)

1 -4

5 1 + 38l/a1,

(Jra' b'1)'(u1) = 0.

Then a subsequence of (ul) converges to a critical point of Pa'b) with critical value 1 by Lemma 8.11 and hence (a, b) e E(Ap, Bp +, B;) by Lemma 8.10, a contradiction.

PROOF OF THEOREM 8.13. Let 6 > 0 be as in Lemma 8.14 and let (a', b') e B5(a, b). Then IJPa'b)(u)

- JPa'b)(a) < la' - al JJ (n) + Ib' - bI Jp (u) Vu e M

< 6 Jp(u) < 8/A1

by Theorem 4.6 (ii), so (fla'b')) (JP(" .b)), D

1+a/a, D (1Pa'b))1+2b/a,

.

(JPa'b ))1+3b/a,.

Since JPa'b) has no critical values in [1, 1 + 28/Al], there is a deformation retraction 77: (Jpa'b))1 x [0, 1] -' (fla'b))1

of (JPa'b))1 onto (JPa'6) )1+25/A, by Lemma 3.47 applied to -Jpab) Similarly,

there is a deformation retraction 7 (,7p(, b'))1+d/a, x [0, 1]

-' (Jla'b'))1+d a,

of (Jp° b))1+s/a, onto (JPa''b')1+3s/a,. The map (JPa'b))1 x [0, 1]

(u, t)

-' (JPa'b))1, 77(u,2t),

0 < t < 1/2

rj (,q(u, 1), 2t - 1),

1/2

u for a subsequence by (A4). Now passing

to the limit in Apuj =

B(a,b)

aj +

'(uj) -g(uj)

3 gives Ap u =

BP(alb)

u, and u # 0 since IIuII = 1.

For R > 0, let

UR=RIM={ueW:Ip(u)0

8.4. PERTURBATIONS AND SOLVABILITY

107

by Lemma 3.3, and then =6RP-1,

uEn.t II(a6)(Ru)II

R>0

by homogeneity. It follows that (8.19)

as R -.oc

inf

UGM

since sup Ig(RM)II* = o(RP-1) by (8.17). Take a smooth function (p [0, co) [0,1] such that o = 1 on [0, 1] and co = 0 on [2P, cc) and set 'PR(u) = 4)(a,b) (u) + w(IP(u)/RP) G(u).

Then (i) is clear. Since jj(co(Ip/RP))'(Ru)fl* = Icp'(IP(Ru)/RP)IIIAP(Ru)11*/RP

ueM

= O(R-1),

by (1.5) and supIG(RM)I = o(RP) by (8.18), (8.19) holds with 4) replaced

by 'FR also. So for sufficiently large R, inf l1-D'(u)ll* > 0, (8.20)

inf II''R(u)II* > 0 ueU`f

UGUn`

and hence (ii) follows. O

By (8.20), every (PS) sequence for 'FR has a subsequence in UR, which then is a (PS) sequence of 4) by (i) and hence has a convergent subsequence. Since 'F(a b) and G are bounded on bounded sets and V is bounded, 'FR

is also bounded on bounded sets. By (ii), the critical values of 'FR are bounded from below by inf 'FR(UR). Taking the a in (3.25) to be less than both inf 'FR(U2R) and inf 'F(a b) (U2R), say a', gives CQ((DR, 00) = IIQ(W, 4>R)

= H (W,

,p(a,b))

since 'R and (P(a b) lie outside U2R, where (DR = '(a,6) by (2). Since the origin is the only critical point Of 'F(a b) and a' < (a b) (0), FI4(W, '(a b))

G"1(D(a b), 0)

by Proposition 3.12.

The main result of this section is

Theorem 8.20. Assume (A1) - (A4), (B1) - (B3), (Bl) - (B4), and (8.16) and (8.17). If (a, b) is in a path component of R2\E(AP , By , By) containing a point of the form (A, A), then (1.1) has a solution. PROOF. It suffices to show that the functional 'FR in Lemma 8.19 has a critical point. Since C9(4)R, co)

C9(4, (a,b), 0) # 0

for some q by Corollary 8.15, this follows from Proposition 3.16.

CHAPTER 9

Indefinite Eigenvalue Problems In this chapter we drop the condition (B2) in the eigenvalue problem (4.1) and allow Jp to change sign. Then the eigenvalues may be positive or negative, but not zero since u = 0 is the only solution of Ap u = 0 by (A2).

9.1. Positive Eigenvalues First suppose

(B2) the set

M+:= {ueM:J4(n)>0} is nonempty. Since Jp is continuous and even, M+ is a symmetric open submanifold of M. As in Section 4.1, the positive eigenvalues and the associated eigenvectors on M+ are the critical values and the critical points of the positive and even Cl-functional

TIM+_

1

Since Jp is bounded on the bounded set M+ by Proposition 1.1, '+ is bounded from below by a positive constant. If (ui) c M+ is a (PS), sequence of W+, then c 0, and the argument in the proof of Lemma 4.5 shows that c > 0 and that a renamed subsequence

c, then Jp(u) = 1/c > 0 converges to some it e M. Since 1/Jp(uj) So ,p+ satisfies (PS). In particular, the set Ex of and hence it e eigenvectors of A that lie on M+ is compact and the positive spectrum M+.

o+(Ap , Bp) consisting of all positive eigenvalues is closed. We also have the following deformation lemma for ,y+ even though M+ is not complete.

Lemma 9.1. If c, b > 0, then there are e > 0 and 77 e C([0, 1] x .A4+, M+), with 77(t, ) odd for all t e [0,1], satisfying (i) d(a7(t, u), u) < bt V(t, u) e [0, 1] x M+, (ii) 4`+(77(t, u)) Y+ (u) V(t, u) e [0, 1] x M+, (iii) 77(1, (iD+)c+e\N5(Kc)) e ( +)c-e

PROOF. We apply Lemma 3.51 to 4) = - JpjM at the level a = -1/c. Its critical set at this level is also K° by Proposition 3.54. An argument similar to that in the proof of Lemma 4.5 shows that 4) satisfies (PS)z. Thus, there 109

9. INDEFINITE EIGENVALUE PROBLEMS

110

are e > 0 and

e c([0, 1] x M, M), with (t, ) odd for all t e [0,1],

satisfying

u), u) < St V(t, u) e [0,1] x M,

(9.1) (9.2)

ID (

(t, u)) < ID (u)

(9.3)

V(t, U) E [0,1] x M,

X1(1,(V+i\N6(K`))

IfueM+, then JJ(i (t, u)) > JP(U) > 0 by (9.2), so r1(t, u) e M+. Let czF (9.4)

1+cE'71=17I[0,1]xM+

E

Then (i), (ii), and (iii) follow from (9.1), (9.4), and (9.3), respectively.

Let d+ = i(M+). Then 1 < d+ < d by (BZ ), (i1), (i2), and (i8). Let

K+={kEN:k A'), then

Cq(4)a,0) ^ Hq-1(( )a) (resp. Hq In particular, CO(4)a, 0) = 0 if A < A

+q or A > All

9.4. CRITICAL GROUPS OF PERTURBED PROBLEMS

(iii) Ifak+1 < A < A

or A' < A < Ak

1,

113

then

Cti((Da,0) # 0.

In particular, Corollary 4.10 holds without (B2). We emphasize that Theorem 9.6 applies in all possible cases: 0 is sufficiently small, and apply Proposition 3.34.

Case 1: ak < A < ak+. Let A0 = ( +)Tk and Bo = (`I'+)ak+l u (M\M+). Theorem 9.2 (iii) gives (3.32) with q = k, and (3.33) follows for sufficiently small r since A

H-1

-1+0(1))

,

7rM(u) E

k

(9.13)

IP(u)

1 - + +o(1),

9t4(u) E (F+)ak+l

k+1

7rM(u) e M\M+

1 + 0(1),

as u ->0 by (9.11). Let Ao = (Y )Ak and Bo = (T-) k+1 U

Case 2: Ak+1 < A < Ak.

(M\M-). Theorem 9.4 (iii) gives (3.32) with q = k, and (3.33) follows for sufficiently small r since 1

A - 1+0(l) /f

\Ak

(9.14)

A

+o(1),

,

7rM (u) E ( 7rM(u) E (r

k+1

+o(1), as u -. 0 by (9.11).

7r-M(u) E M\M-

9.4. CRITICAL GROUPS OF PERTURBED PROBLEMS

115

Now suppose (9.15)

F(u) = A Jp(u) + o(IIuIIP) as

co.

IIuII

We have

Proposition 9.10. Assume (Al) - (A4), (B1), (B3), (9.15), and 4) satisfies (C).

(i) If Aj < A < A', then Cg (4), oo)

Sgo Z2.

(ii) If ak+1 0 such that (Ap;ui, ui)i

Co IIuiIIP'

(Ai3) a potential operator, 117

Vu i e Wi,

10. ANISOTROPIC SYSTEMS

118

AP satisfies (A4), and F E C1(W, Ilk), F(0) = 0, and F' = (FF.,... , Fu,,,)

W W* is compact. As in Chapter 1, the potential IP of AP satisfying IP(0) = 0 is given by (10.2)

IP(u) = LJ

Pi

(AP:ui,'ui)i,

F can be written as F(u)

1

J0

(Fu; (tu), ui)i dt,

and the variational functional associated with equation (10.1) is

41(u) = IP(u) - F(u) 1

M

_

1

(APiui,

[Pi

ui)i

- J0 (Fui(tu), ui)i dtI

,

u E W.

Although not homogeneous, IP is even and satisfies (10.3)

.

co

1 Iluilir'

IIuiIIP' -< IP(-) -< Co

2=1 Pt

Vu E W

T=1 Px

for some Co >, co. Writing (u1/p1,. .. , un/pm) = u/p, we also have (10.4)

(IP(u), u/P) = (AP u, u/P) = IP(u),

analogous to (4.3).

Proposition 10.1. If each Wi is uniformly convex and Pi-1Il Pi < villi , ( APiua, u i)i = ri Iu i i (A Pt ui,vi)i - ri Iluilli

i, Vu z,

i

E Wi

for some ri > 0, then AP satisfies (A4).

PROOF. If uj - u and (AP uj, u? - u) m

0, then

Pi-1

(u

0

ti=1

uilli)

\AP'uj", ut

/i-

\APiua, ui/ i

- (APiui, ui ) + (APiui, ui)i )

_ (APu7,u7 - u) - (APu,ui -u) -, 0, so, for each i,

luti II

luilli and hence acs

ui by uniform convexity.

Example 10.2. Consider the p-Laplacian system

JDP u = Vu F(x, u) (10.5)

u=0

in 52

on8Q

10.1. EIGENVALUE PROBLEMS

119

where p = (p1, ... , pm) with each pi E (1, oo), u = (ui, ... , um), Ap u = (Ap,ui,..., Apmu.m), and F e Cl(S2 x W') satisfies F(x, 0) e 0 and OF

(10.6)

fVLi

0} is nonempty. Since J is continuous and even, M+ is a symmetric open submanifold of M. We will show that the positive eigenvalues and the

10. ANISOTROPIC SYSTEMS

122

associated eigenvectors on M+ are the critical values and the critical points

of the positive and even Cl-functional

J M+ Since J is bounded on the bounded set M+, ,p+ is bounded from below by a positive constant. Since

(') (u) =

)2 =

II(u) = Apu,

+(u)2 J'(u),

it follows from Proposition 3.54 that the norm of ( +)'(u) e T, *M is given by

I('I`+)'(u)Ilw = min IltApu+lY+(u)2 J'(u)II".

(10.15)

jLrR

Lemma 10.7. Positive eigenvalues of (10.12) coincide with the critical values of 4l+, i.e., A > 0 is an eigenvalue if and only if there is a u e M+ such

that (W+)'(u) = 0 and i+(u) = A. PROOF. By (10.15), (lY+)'(u) = 0 if and only if (10.16) for some ju a R. If (10.16) holds, applying it to u/p and using (10.4) and

Proposition 10.4 (iv) gives

J(u) _ II(u)

_ +(u)2 (J'(u), u/P) = (Ap u,u/P)

0,

so (10.16) reduces to (10.12) with A = +(u). Conversely, if A > 0 is an eigenvalue and u e M+ is an associated eigenvector, A _ (Apu,u/P) = IP(u) =

(J'(u),u/P)

J(u)

All

+(u),

so (10.12) implies (10.16) with it = -lI+(u).

Example 10.8. In Example 10.5, ,n

M={(u1i...,um)eWo,Pl(52 )x...xWo,Pm(52):E J Vuilp'=1 l

i_1 Pi

4' + (u)

=

1

J V(x) lulIrl ... In4rm Lemma 10.9. + satisfies (PS).

12

11

10.1. EIGENVALUE PROBLEMS

123

PROOF. Let (ui) c M+ be a (PS), sequence. Then c > 0. By (10.14), (ui) is bounded, so a renamed subsequence converges weakly to some u e W

since W is reflexive, and J'(u3) converges to some L e W* for a further subsequence since J' is compact. Since the sequence (uj/p) is also bounded and converges weakly to u/p, then

T+(ui) =

I

1

us/p)

(L,u/p) by Proposition 10.4 (iv) and Lemma 3.4, so c > 0. By (10.15), (10.17)

t 3 Ap u' + +(u))2 J'(uj) - 0

for some sequence (p3) c R. Applying (10.17) to ui/p and using (10.4) and Proposition 10.4 (iv) gives lay +W+(uj) -. 0, so µi - -c # 0. Now applying

0 since (J'(u-'),ud - u) -. 0 by (10.17) to uu - it gives (Ap ui,ui - u u e M for a subsequence by (A4). Since 1/J(ug) - c, Lemma 3.4, so uj 0 then J(u) = 1/c > 0 and hence u e M+. Lemma 10.9 implies that the set Ea of eigenvectors of A that lie on M+ is compact and the positive spectrum a+(Ap, J') consisting of all positive eigenvalues is closed. Applying Lemma 3.51 to

JIM at the level a= -1/c, and noting

that an argument similar to that in the proof of Lemma 10.9 shows that satisfies (PS)a, we see that Lemma 9.1 still holds. So, defining K+ and ak by (9.5), (9.6), and (9.7), we have the following theorem (the second equality in (10.18) follows from (10.10) and (10.11)).

Theorem 10.10. Assume (Ail) - (Ai3), (A4), and J e ,7 with J' compact. Then (ak )kEK+ is a nondecreasing sequence of positive eigenvalues of (10.12).

A, then i(EA) > in. In particular, there = are d+ distinct pairs of eigenvectors on (ii) The smallest positive eigenvalue is

(i) If At =

M+.

(10.18)

(iii) We have

A = umin

+(u) = min J(ua) > 0. + a a#0

k

i((I+)A) =

(iv) If d+ = co, then ak J' +co. We now consider the case where the set

M- := {u E M : J(u) < 0}

If At < A < ak+1, then

k ) = k.

10. ANISOTROPIC SYSTEMS

124

is nonempty. Applying the preceding discussion to -J as in Chapter 9, we see that M- is also a symmetric open submanifold of M and that the negative eigenvalues and the associated eigenvectors of (10.12) are the critical values and the critical points of the negative and even C1-functional ip- _

1

J M_

which is bounded from above by a negative constant and satisfies (PS). The set Ea of eigenvectors of A that lie on M- is compact and the negative spectrum a- (AI,, J') consisting of all negative eigenvalues is closed. Defining 1C- and ak by (9.8), (9.9), and (9.10), we have

Theorem 10.11. Assume (Ail) - (At3), (A4), and J e ,7 with J' compact. Then (ak)kEK- is a nonincreasing sequence of negative eigenvalues of (10.12).

(i) If Ak = ..' = Ak+m-i = A, then i(E),) >, m. In particular, there

are d- distinct pairs of eigenvectors on M-.

(ii) The largest negative cigenvalue is

Al = max Y (u) = max IP(ua) < 0. uEM-

uEM- J(ua) a#0

(iii) We have i(M \('I )ak) < k

)ak ). If Ak+l < A < Ak, then

i(('Y )ak) = i(M \('I` )A) = i(('I )a) = i(M \(y )ak+) = k. (iv) If d- = cc, then Ak \ -oo. Since u = 0 is the only solution of Ap u = 0 by (A%2), 0 0 o(Ap , J') and hence

a+(Ap , J') u a (Ap, J') = a(Ap, J'). When A 0 a(Ap, J'), the critical groups of the associated functional

Fx(u) = Ip(u) - A J(u),

ucW

at zero are given by (4.11).

By (10.10) and (10.11),

'Pa(ua) = lal 'Pa(u), so O(4') c 4)0. Hence 4150 contracts to {0} via (u, t)

4)°x [0,1] -* 4' ,

ul_t.

If u A 0, au := Ip(u) > 0 by (10.3) and the path component of O(u) containing u intersects M at u := ua-1 since Ip(u.) = 1 by (10.10). Hence

$a\ {0} deformation retracts to 415a n M via {0}) x [0, 11 -* $O\ {0}

,

(u, t) '-' ul-t+tau'

Thus, (4.13) still holds, and as in Chapter 9 we get

10.2. CRITICAL GROUPS OF PERTURBED SYSTEMS

125

Theorem 10.12. Assume (Ail) - (Ai3), (A4), J e 9 with J' compact, and A

a(A,,J'). (i) If \- 1/Aj) for u e M+ (resp. M-) by Theorem 10.10 (ii) (resp. Theorem 10.11 (ii)),

A Al (10.24)

ueM+,A>0

+0(1)

4)(ua) % a( I - a +o(l) )

u e M-, A < 0

1

a (1 + o(1)),

otherwise

by (10.23), and (10.21) follows from Proposition 3.18.

(ii) We show that 4) has a cohomological local splitting near zero in dimension k, with U = U, and h = hr where r > 0 is sufficiently small, and apply Proposition 3.34. Note that

A=h,.(IAo)={ua:ueAo,05acr}, B=h,.(IBo)u{0}={ua:ueBo,0