CRITICAL POINT THEORY AND ITS APPLICATIONS

CRITICAL POINT THEORY AND ITS APPLICATIONS

By

WENMING ZOU Tsinghua University, Beijing, China

MARTIN SCHECHTER University of California, Irvine, California, USA

^ Spri ringer

Library of Congress Control Number: 2006921852

ISBN-10: 0-387-32965-X e-ISBN: 0-387-32968-4

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

Printed on acid-free paper.

AMS Subject Classifications: 35J50, 58E05, 47J30, 49505, 58E30

© 2006 Springer Sciencen-Business Media, LLC

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America.

9 8 7 6 5 4 3 2 1

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W. Zou dedicates this book to his parents: LIANG-SHENG ZOU & GUO-XIU ZENG.

M. Schechter dedicates this book to his wife, children and grandchildren (currently 22) and great grandchildren (currently one).

Contents

Preface ix

1 Preliminaries 1 1.1 Partition of Unity in Metric Spaces 1 1.2 Sobolev Spaces 2 1.3 Differentiable Functionals 7 1.4 Topological Degrees 13 1.5 An ODE in Banach Space 16 1.6 The (PS) Conditions 19 1.7 Weak Solutions 20

2 Functionals Bounded Below 25 2.1 Pseudo-Gradients 25 2.2 Bounded Minimizing Sequences 26 2.3 An Application 31

3 Even Functionals 37 3.1 Abstract Theorems 37 3.2 High Energy Solutions 44 3.3 Smah Energy Solutions 49

4 Linking and Homoclinic Type Solutions 55 4.1 A Weak Linking Theorem 55 4.2 Homoclinic Orbits of Hamiltonian Systems 64 4.3 Asymptotically Linear Schrodinger Equations 73 4.4 Schrodinger Equations with 0 G Spectrum 74 4.5 The Case of Critical Sobolev Exponents 89 4.6 Schrodinger Systems 101

4.6.1 The Superlinear Case 102 4.6.2 The Asymptotically Linear Case 112

viii CONTENTS

5 Double Linking Theorems 117 5.1 A Double Linking 117 5.2 Twin Critical Points 120 5.3 Eigenvalue Problems 129 5.4 Jumping Nonlinearities 131

6 Superlinear Problems 141 6.1 Introduction 141 6.2 Proofs 145 6.3 The Eigenvalue Problem 152

7 Systems with Hamiltonian Potentials 159 7.1 A Linking Theorem 159 7.2 Hamiltonian Elliptic Systems 167

8 Linking and Elliptic Systems 179 8.1 An Infinite-Dimensional Linking Theorem 179 8.2 Elliptic Systems 187

9 Sign-Changing Solutions 195 9.1 Linking and Sign-Changing Solutions 195 9.2 Free Jumping Nonlinearities 202

10 Cohomology Groups 215 10.1 The Kryszewski-Szulkin Theory 215 10.2 Morse Inequalities 227 10.3 The Shifting Theorem 231 10.4 Critical Groups of Local Linking 242 10.5 Computations of Cohomology Groups 244 10.6 Hamiltonian Systems 253 10.7 Asymptotically Linear Beam Equations 269

Bibliography 287

Index 317

Preface

Since the birth of the Calculus of Variations, it has been realized tha t when they apply, variational methods can obtain bet ter results than most other methods. Moreover, they apply in a very large number of situations. It was realized many years ago tha t the solutions of a great number of problems are in effect critical points of functionals.

In this volume we present some of the latest research in the area of critical point theory. Many new results have been recently obtained by researchers using this approach, and in most cases comparable results have not been ob­tained by other methods. We describe these methods and present the newest applications.

In a typical application, one first establishes tha t the solution of a given problem is a critical point of a functional G{u) on an appropriate space, i.e., a "point" in the space where G\u) = 0. Finding the points where the derivatives vanish is tan tamount to solving the problem. The main difficulty is finding candidates. In this connection, one can use "geometrical" considerations. But geometrical considerations do not involve derivatives, and usually the best they can produce are Palais-Smale sequences, i.e., sequences of the form

G{uk) -^ a, G\uk) -^ 0.

The existence of such a sequence is not enough to produce a critical point. It is possible tha t such a sequence is converging to infinity. However, if one can show tha t the sequence has a convergent subsequence, then one indeed obtains a critical point. A functional tha t has the property tha t every Palais-Smale sequence for it produces a critical point is said to satisfy the Palais-Smale condition.

Wha t is one to do if the corresponding functional does not satisfy the Palais-Smale condition? In the present volume, one of the purposes is to consider just this situation. The trick here is to find bounded Palais-Smale sequences directly from the linking geometry. In most cases such sequences

X PREFACE

produce critical points. One might think that such methods are severely restricted. However, the number of such methods and applications found here should convince anyone otherwise. It is surprising that so much has been accomplished under this handicap resulting in new variational methods. Another purpose of this book is a description of the so-called topological method. We present a new Morse theory which satisfactorily fits strongly indefinite functionals.

We include such topics as extrema, even functionals, weak and double link­ing, sign-changing solutions, Morse inequalities, and cohomology groups. The applications we describe include Hamiltonian systems, Schrodinger equations and systems, jumping nonlinear it ies, elliptic equations and systems, superlin-ear problems and beam equations.

The book is organized as follows.

In Chapter 1, we provide some prerequisites for this monograph. We collect some knowledge of degree theory, Sobolev space and so forth. Basically, these theories are essentially known and readily available in many books. Well-trained readers may skip this chapter.

In Chapter 2, we present some theorems concerning functionals which are bounded below on Banach spaces or Finsler manifolds.

Chapter 3 is devote to critical point theory on even functionals. Some variants of the fountain theorem will be established without (PS) type as­sumptions. Applications to Schrodinger equations and Dirichlet boundary value problems will be given. We will show readers how to get infinitely many solutions.

In Chapter 4, we establish a weak infinite-dimensional linking theorem. It not only unifies the classical results but also gives us more information. This abstract theory works perfectly for some PDEs and ODEs with pure continuous spectrum. Therefore, applications will be considered mainly on homoclinic type solutions of asymptotically linear Hamiltonian systems and Schrodinger equations, superlinear Schrodinger equations with zero as a point of the spectrum or with critical Sobolev exponents. In particular, Schrodinger systems depending on time will be discussed.

Chapter 5 concerns twin critical points resulting from double linking. Roughly speaking, if A links 5 , does B link A? Can they yield two different critical points without the (PS) type compactness conditions? We will give positive answers. Applications on eigenvalue problems and Dirichlet elliptic equations with jumping nonlinearities will be studied.

PREFACE xi

In Chapter 6, we solve elliptic semilinear boundary value problems in which the nonlinear terms are quite weak super-linear. That is, the nonlinearities need not satisfy a superquadracity condition of the Ambrosetti-Rabinowitz type. Because of this, we are able to include more equations than hitherto permitted. Some new tricks will be seen.

In Chapter 7, we assume that A links B. Let Bi and B2 be two linear bounded invertible operators. We describe the situation in which the values of the functional H are separated by BiB and B2A. Then BiB and B2A become much more complicated. We prove the existence of a critical point of H without assuming (PS) type conditions. This theory is applied to some special elliptic systems.

In Chapter 8 we prove an infinite-dimensional linking theorem and apply it to elliptic systems with gradient type potentials.

In Chapter 9, we study the existence of sign-changing solutions for non­linear elliptic equations via linking methods. A linking type theorem is es­tablished with the location of the critical point in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities. Under stronger conditions, we show that the existence of sign-changing critical points can be independent of the Fucik spectrum which usually is indispensable for such cases.

In Chapter 10, a more advanced Morse theory will be introduced. We first present the W. Kryszewski-A. Szulkin infinite-dimensional cohomology theory and a new Morse theory associated with it. Then we develop some methods of computing the cohomology critical groups precisely. Applications to Hamiltonian systems and beam equations will be considered.

The present monograph is based on results obtained by ourselves or through direct cooperation with other mathematicians such as S. Li, A. Szulkin, Z. Q. Wang and M. Willem. It is not intended to be complete.

The materials covered in this book are presented at a level suitable for advanced graduates and Ph. D. students following the development of new results, or anyone who wishes to seek an introduction to critical point theory and the study of differential equations by variational and topological methods. The chapters are designed to be as self-contained as possible.

xii PREFACE

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

Wenming Zou

Tsinghua University, Beijing

Martin Schechter

University of California at Irvine

Chapter 1

Preliminaries

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

1.1 Partition of Unity in Metric Spaces

Assume {E,d) is a metric space with a distance function (i(-, •). Let A C E and let 11 be a family of open subsets of ^ . If each point of A belongs to at least one member of 11, then 11 is called an open covering of A.

Definition 1.1. Assume that H is an open covering of a subset A of E, then n is called locally finite if for any u ^ A, there is an open neighborhood U such that u e U and that U intersects only finitely many elements of li.

The following result is due to A. H. Stone [347].

Proposition 1.2. Any metric space {E,d) is paracompact in the sense that every open covering li of E has an open, locally finite refinement Q, i.e., 6 is a locally finite covering of E and for any Vi of Q, we can find a Ui of li such that Vi C Ui.

Proposition 1.3. Assume that {E, d) is a metric space with an open covering n . Then li admits a locally finite partition of unity {\i}i^j subordinate to it satisfying:

(1) Xi : E ^ [0,1] is Lipschitz continuous;

(2) {Vi}i^j is a locally finite covering of E, where Vi = {u ^ E : \i{u) ^ 0}, J is the index set;

2 CHAPTER 1. PRELIMINARIES

(3) for each Vi, there is a Ui eli such that Vi C Ui]

(4) E i e J^ iW = l ' V u e ^ .

Proof. Since {E, d) is a metric space with an open covering 11, by Proposition 1.2, there is an open, locally finite refinement B, i.e., B is locally finite and for any Vi of B, we can find a [/ of 11 such that Vi ^Ui. We define

pi{u) = d{u,E\Vi), i G J.

Then pi is locally Lipschitz. Let

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

1.2 Sobolev Spaces

Let O be an open subset of R ^ , AT G N. Define

1/^(0) := {i : O ^ R is Lebesgue measurable, ||I^||LP(Q) < oo},

where

\\U\\LP{Q) = y \u\Pdxj , l < p < + o o .

If p = +00,

\u\\LOOm\ = ess sup \u\ := inf sup |i^| Q Acfl,meas{A)=0 Q\j^

where meas denotes the Lebesgue measure. Let

^ L ( ^ ) •={u:n^Il,ue LP{V) for each V CC O},

where y c c O < ^ y c y c O and V is compact. We will in this book denote \MLP{Q) by ll^llp or \u\p.

Let C^{fl) denote the space of infinitely differentiable functions (/> : O ^ R with compact support in O. For each (j) G C^(^) and a multiindex a = (o^i,..., Q AT) with order \a\ = ai -\-... -\- ajy, we denote

dx'^' • • • dx'}" i-jV

uD'^(j)dx = (-1)1^1 / v(j)dx

1.2. SOBOLEV SPACES 3

Definition 1.4. Suppose u^v e Ll^^{ft). We say thatv is the a^^-weak partial derivative of u, written D^u = v provided

for aU(l)eC^{n).

It is easy to check that the o^^ -weak partial derivative of u, if it exists, is uniquely defined up to a set of measure zero.

Let C^{ft) be the set of functions having derivatives of order < m being continuous in Q {m = integer > 0 or m = oo).

Let C"^(0) be the set of functions in C"^(0) all of whose derivatives of order < m have continuous extension to fl.

Definition 1.5. Fix p G [1, +oo] and A: G N U {0}. The Sobolev space

consists of all u : Q ^ H which have a^^- weak partial derivatives D^u for each multiindex a with |Q | < A: and D^u G 1/^(0).

If p = 2, we usually write

H^{n) = iy^'2(0), A: = 0 ,1 ,2 , . . . .

Note that H^{n) = L'^{n). We henceforth identify functions in ly^'^(O) which agree a.e.

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

i/p ( / \ vjv

Definition 1.7. We denote

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

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

CHAPTER 1. PRELIMINARIES

The following results can be found in L. C. Evans [147].

Proposi t ion 1.8. For each A: = 1, 2 , . . . and I < p < +oo, the Sobolev space

f VF^'^(O), II • ||vi/fc'P(Q)) -5 0. Banach space and so is WQ'^{Q). In particular,

H^{Q),HQ{Q) are Hilhert spaces.

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

Definition 1.10. A function u : Q C R ^ ^ H is Holder continuous with exponent 7 > 0 i/

1(7) .- \u(x) — u(y)\ sup ^ ^ ^ < 00.

\x-y\-r

Definition 1.11. The Holder space C' '' (f2) consists of all functions u G C''(f2) for which the norm

(y.\=k

\\u\\cK.m:= Y, P"«llc(n) \a\<k

is finite. It is a Banach space. We set C^'^(O) = C^{Q).

We have the following imbedding results, see R. A. Adams [4], L. C. Evans [147] and D. Gilbarg-N.S. Trudinger [174].

Proposi t ion 1.12. If Q is a bounded domain in H^, then

rk,p <'^(0)

L^(0), kp<N,l<q< Np/{N - kp),

C"^'^(0), 0<a<k-m- N/p and

0 < m < k- N/p < m + 1.

Proposi t ion 1.13. If ft is a bounded domain in H^, then

<'^(0)

L^(0), kp<N,l<q< Np/{N - kp),

C"^'^(0), {)<a<k-m-N/p and

0 < m < k — N/p < m -\-1.

1.2. SOBOLEV SPACES 5

The following proposition can be found in H. Brezis [64] and M. Willem [377].

Proposition 1.14. The following imbeddings are continuous:

H\'R^)^LP(R^), if 2<p<oo,N = l,2,

H^K^) - ^ L ^ ( R ^ ) , if 2<p<2%N>3,

where 2* := 2N/{N-2) ifN>3 and 2* = +oo if N = 1,2, is called a critical exponent.

For AT > 3, let

be the best Sobolev constant. Then, by G. Talenti's results (cf. [362]),

o_l |v^lli \\u\\l,'

where

(Ar-2)/4 ( iV(iV-2))

U{x) = ^ ' (N-2)/2

Note that if R ^ is replaced by a bounded domain, S is never achieved. Let

iVC(a;)e(^-2)/2 ^e{x) :--

(£2 + Ixp) ( A f - 2 ) / 2 '

where A = {N{N - 2))^^ ^^^".e > 0 and ^ e Cg°(R^, [0,1]) with i{x) = 1 if |a:| < r /2; ^(x) = 0 if |x| > r, where r can be chosen to meet different requirements.

Proposition 1.15. The following estimates are true (see e.g. pp.35 and 52 ofM. Willem [377]):

r ce2|^n£|+0(e2), A = 4,

ll^e||i =

where c > 0 is a constant.

6 CHAPTER 1. PRELIMINARIES

We shall frequently use the following Gagliardo-Nirenberg Inequality, see L. Nirenberg[262] (see also L. C. Evans [147] and J. Chabrowski [89]).

Proposition 1.16. For every u G H^{¥i^)^

\\u\\,<c\\Vu\mu\\l--

N N -2 N with — = a ^ (1 ~ ^) — 5' ^ l?^^ ^ [O5II5 where c is a constant

p 2 r depending on p,a, r, N.

The following concentration-compactness lemma due to P. L. Lions [228] is also a powerful tool in dealing with the Schrodinger equation. Lemma 1.17. Let r > 0 and q G [2,2*). For any bounded sequence {u^} of

E := HHR^), if

^P / \un\^dx ^ 0 , n ^ 00, R ^ JBiy^r)

sup yGR^ JB{y,r)

where B{y^r) := {u e E : \\u — y\\ < r}, then Un ^ 0 in I/^(R^) for q<p<2\

Proof. We just consider A > 3. Choose Pi,P2,^ > l,t^ > 1 such that Pit = q^p2t' = 2*,l/ t + 1/t' = l ,pi + p2 = P' By Holder's Inequality and Proposition L13, we have

\Un\^dx B{y,r)

<( [ lur^r'dxY^'f [ lur^r^'dxY^' ^JB{y,r) ^ ^JB{y,r) ^

<-^\, ' " lB{y,r)

< c ( • • - - - ' ^ ' ^ ^ - - - ' ' ' ' '

lB{y,r) ^ ^JB{y,r)

(In this book, the letter c will be indiscriminately used to denote various constants when the exact values are irrelevant.) Covering R ^ by balls with radius r in such a way that each point of R ^ is contained in at most A + 1 balls, then we have

I

\Unr*dxy"( / {ul + \SJUn\'')dxy''\

/ |u„|^dx<(A^ + l)c sup f / \un\'^dx) 1/t

which impUes the conclusion. This proves the theorem. D

1.3. DIFFERENTIABLE FUNCTIONALS 7

1.3 Differentiable Functionals

Let ^ be a Banach space with norm || • ||. Let U C E he SLU open set of E. The dual (or conjugate) space of E is denoted by E^, i.e., E^ denotes the set of bounded linear functionals on E. Consider a functional I : U ^ H.

Definition 1.18. The functional I has a Frechet derivative F ^ E' at u ^ U

heE,h^o \\h\\

We define r{u) = F or \/I{u) = F and sometimes refer to it as the gradient of / at u. Usually, r{') is a nonlinear operator. We use C^(t/, R) to denote the set of all functionals which have continuous Frechet derivative on U. A point u ^ U is called a critical point of a functional / G C^{U, R), if

r{u) = 0.

Definition 1.19. The functional I has a Gateaux derivative G ^ E' atu ^U if, for every h ^ E,

t^o t ^ ^

The Gateaux derivative ai u ^ U is denoted by DI{u). Obviously, if / has a Frechet derivative F e E^ dit u e U, then / has a Gateaux derivative G e E^ dit u and I\u) = DI{u). But the converse is not true. However, if / has Gateaux derivatives at every point of some neighborhood oi u ^U such that DI{u) is continuous at u^ then / has a Frechet derivative and I'{u) = DI{u). This is a straightforward consequence of the Mean Value Theorem.

Let f{x,t) be a function on O x R, where O is either bounded or un­bounded. We say that / is a Caratheodory function if / (x , t) is continuous in t for a.e. x G O and measurable in x for every t G R.

Lemma 1.20. Assume p > l^q > I. Let f{x,t) be a Caratheodory function on ft X H and satisfy

\f{x,t)\ < a + 6|t|^/^ V(x,t) G O X R ,

where a, 6 > 0 and O is either bounded or unbounded. Define a Caratheodory operator by

Bu:= f{x,u{x)), ueLP{n).

Let {i^fcj^o ^ LP{Q). If\\uk -uoWp -^ 0 as k ^ +oo, then \\Buk - Buo\\q -^ 0 as k ^ oo. In particular, if ft is bounded, then B is a continuous and bounded mapping from 1/^(0) to L^{ft) and the same conclusion is true if ft is unbounded and a = 0.

8 CHAPTER 1. PRELIMINARIES

Proof. Note that there is a renamed subsequence such that

(1.1) Uk{x) -^ uo{x), a.e. x e ft.

Since / is a Caratheodory function,

(1.2) Buk{x) -^ Buo{x), a.e. x e ft.

Let

(1.3) Vk{x) := a + 6|^fe(x)|^/^ A: = 0,1, 2 , . . . .

Then by (1.1)-(1.3),

(1.4) \Buk{x)\ < Vk{x) for all x ^ ft; Vk{x) -^ vo{x) a.e. x ^ ft.

Since \uk\^ -\- \UO\P — \\uk\^ — |i^o|^| ^ 0, by Fatou's Theorem, we have

/ hminf (\ukf + i^or - ii^^r - i^ori)^^ (1.5) < hminf / (\ukf^\uof-\\ukf-\uof\)dx.

Combining (1.1)-(1.5), we see that

(1.6) lim / | |^fe |^- |^orM^ = 0.

It follows that

(1.7) / \v^-v^\Ux<h^ \ \\u^\^-\u^\^\dx^^

as A: ^ oo. Since there are constants C > 0, Ci > 0 such that

\Buk-Buo\i < C{\Buk\'^ + \Buon

< C,i\vk-vo\^ + \von

a.e. X G O, by Fatou's Theorem,

/ liminf (Ci{\vk - ^ol' + l^ol") - \Buk - Buo\'')dx

(1.8) < liminf / (Ci{\vk-vo\''^\von-\Buk-Buo\Adx.

1.3. DIFFERENTIABLE FUNCTIONALS 9

By (1.2), (1.3), (1.7) and (1.8), we have

\\Buk-Bu\\q^O.

Finally, if O is bounded, then for any u G 1/^(0), evidently we have

(1.9) \\Bu\U<c + c\\urJ'^,

where c > 0 is a constant. Inequality (1.9) remains true if O is unbounded and a = 0. Therefore, 5 is a continuous and bounded mapping from 1/^(0) to 1/^(0) and the same conclusion is true if O is unbounded and a = 0. D

The following lemma comes from M. Willem [377].

Lemma 1.21. Assume pi,p2,qi,q2 > 1- Let f{x,t) be a Caratheodory func­tion on O X R and satisfy

\f{x,t)\ < a|t|^i/^i + 6|t| ^/^% \/{x,t) G O X R,

where a,b > 0 and O is either bounded or unbounded. Define a Caratheodory operator by

Bu := / (x , u{x)), uen:= L^' (O) H L^' (O).

Define the space

<£::=L^i(0) + L^2(0)

with a norm

\\u\\s = mf^\\v\\L.iin) + \\w\\L.2in) :u = v^weS,ve L^'{n),w e L^^(O)}.

Then B = Bi -\- B2, where Bi is a bounded and continuous mapping from L^^iQ) to L^^{Q)^i = 1,2. In particular, B is a bounded continuous mapping from Ti to 8.

Proof. Let (f : R ^ [0,1] be a smooth function such that £^{t) = 1 for t G (-1,1); ^(t) = 0 for t ^ ( -2 , 2). Let

^(x,t) =e ( t ) / ( x , t ) , h{x,t) = (1 - e ( t ) ) / ( x , t ) .

We may assume that pi/qi < ^2/^2- Then there are two constants d > 0,m > 0 such that

\g{x,t)\ < (i|t|^i/^S \h{x,t)\ < m|t|^2/^^

10 CHAPTER 1. PRELIMINARIES

Define Biu = g{x,u), u G I/^'(0);

B2U = h{x,u), ueLP^{n).

Then by Lemma 1.20, Bi is a bounded and continuous mapping from L^^^Q) to L^^ (^), ^ = 1, 2. It is readily seen that B := Bi-\-B2 is a bounded continuous mapping from H to S. D

The fohowing theorem and its idea of proof are enough for us to see that the functionals encountered in this book are of C^.

Theorem 1.22. Assume a > 0,p > 0. Let f{x,t) be a Caratheodory function on ft xH satisfying

(1.10) 1/( ,01 < «IC + ^l^r. V(x,t) G O X R,

where a, 6 > 0 and ft is either bounded or unbounded. Define a functional

I{u) := / F{x,u)dx, where F{x,u) = / f{x,s)ds. JQ JO

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

I'{u)h = I f{x,u)hdx, \/h G E. JQ

Moreover, if E ^^^^ L^^'^^E ^^^^ L^^^, then I' : E ^ E' is compact.

Proof. Since E ^^ L^^^(Q) and E ^^ 1/^+^(0), we may find a constant Co > 0 such that

(1.11) ||u;||<,+i<Co||«;||, | |u; | |p+i<Co|k| | , ^weE.

We make use of Young's Inequahty and

(\s\ + \t\Y <2--\\s\- + \tY), T > i , s , t e R .

Combining the assumptions on / , for any 7 G [0,1], it is easy to check that

\f{x,u^-ih)h\ < Ci( |^P+i + l/iP+i + l^l^+i + |/i|^+i),

where Ci is a constant independent of 7. Therefore, for any u^h G E^ by the Mean Value Theorem and Lebesgue's Theorem,

1.3. DIFFERENTIABLE FUNCTIONALS 11

lim I{u + th) - I{u)

(1.12)

lim / f{x,u-\-Oth)hdx

/ f{x,u)hdx

: Fo{u,h),

where 0 G [0,1] depends on u, h, t. Obviously, Fo(i^, h) is linear in h. Further, by (1.11),

\Fo{u,h)\

< I \f{x,u)h\dx

<c\\u\ a + l l k + 1 ll^ll^+lll/^ll.+l)

<c(ii^r + ii^r)ii/iii. It follows that Fo{u,h) is linear and bounded in h. Therefore, DI{u) = Fo(i^, •) G E' is the Gateaux derivative of / at u. Next, we show that DI{u) is continuous in u. Let Bu := f{x^u)^u G E. By Lemma 1.21, B = Bi -\- ^2 , where Bi is bounded and continuous from 1/^+^(0) to I/^^+^^/^(0) and B2 is bounded and continuous from 1/^+^(0) to I/^^+^^/^(0). For any v,h ^ E,

\{DI{u) - DI{v))h\

/ {f{x,u) - f{x,v))hdx Jn

I Jn

{Bu — Bv)hdx

{Biu -\- B2U — Biv — B2v)hdx

< / \Biu - Biv\\h\dx ^ / \B2U - B2v\\h\dx

< Co\\B,U - B,v\\^,^,yjh\\ + Co\\B2U - B2v\\^p^,y4h\\.

This implies that

(1.13) \\DI{u) - DI{V)\\E' <

Co( | |S l^ t -Bl^; | | ( ,+ l ) / , .

where || • \\E' is the norm in E\

\\B2U-B2v\\^p^iyp

12 CHAPTER 1. PRELIMINARIES

Uvk^uinEc L^+i(0) n L^+i(0), then

Therefore, DI{vk) -^ DI{u). This means that DI{u) is continuous in u. Hence, r{u) = DI{u), i.e., / G C^(^,R). Furthermore, if E ^ ^ L^+\ E ^^^^ 1/ + , then any bounded sequence {uk} in ^ has a renamed subse­quence denoted by {uk} which converges to i o in L^^^{Q) and in 1/^+^(0). Hence, 5 i K ) ^ BI{UQ) in L ( ^ + I ) / ^ ( 0 ) ; ^ S K ) ^ ^2(^0) in L ( ^ + I ) / ^ ( 0 ) .

Finahy, DI{uk) -^ DI{uo) in ^^, i.e., r is compact in ^ . This proves the theorem. D

A relation between pointwise convergence of functions and convergence of functionals can be found in H. Brezis-E. Lieb [67].

Sometimes, we wih use the concepts of the second order Frechet and Gateaux derivatives.

Definition 1.23. The functional I G C^{U,Il) has a second order Frechet derivative at u e U if there is an L, which is a linear hounded operator from E to E', such that

r(u^h)-r(u)-Lh ^

We write I"{u) = L.

We say that / G C^(t/, R) if the second order Frechet derivative of / exists and is continuous on U.

Definition 1.24. The functional I G C^{U^Yi) has a second order Gateaux derivative at u ^U if there is an L, which is a linear hounded operator from, E to E', such that

(r{u^th)-r{u)-Lth]v Um - ^— = 0, V/i, veE. t^o t

We write D'^I{u) = L.

Evidently, any second order Frechet derivative of / is a second order Gateaux derivative. Using the Mean Value Theorem, if / has a continuous second order Gateaux derivative on [/, then / G C^(t/, R).

1.4. TOPOLOGICAL DEGREES 13

1.4 Topological Degrees

Since topological degree is an eternal topic of every book on nonlinear func­tional analysis, we just outline the main ideas and results without proofs. Readers may consult the books of L. Nirenberg [263], K. Deimling [131] and E. Zeidler [379].

To construct the degree theory, Sard's Theorem plays a key role. For this, we introduce the following definition.

Definition 1.25. Let U C X := R^ {N > 1) be an open subset and a mapping I G C^{U,X). A point u e U is called a regular point and I{u) is a regular value if I'{u) : X ^^ X is surjective. Otherwise, u is called a critical point and I{u) is the critical value.

Here we state a simplified version of Sard's Theorem. It is from A. Sard [305].

Theorem 1.26. Let U C X := R^ {N > 1) be an open subset and I G C^{U,X). Then the set of all critical values of I has zero Lebesgue measure in X.

Definition 1.27. (Brouwer Degree) LetU C X '.= 11^ {N > 1) be a bounded open subset, I G C^{U,X), p e X\I{dU).

(1) If p is a regular value of I, define the Brouwer Degree by

deg(/, [/,p) := ^ signdet/'(i;),

where det denotes the determinant.

(2) If p is a critical value of I, choose pi to be a regular value (by Sard^s Theorem) such that \\p — Pi\\ < dist{p,I{dU)) and define the Brouwer Degree by

degiI,U,p):=degiI,U,pi).

In item (1), I~^{p) is a finite set when p is a regular value. In item (2), the degree is independent of the choice of pi.

If / G C(f7,X), we may find by Weierstrass's Theorem an approximation of / via a smooth function.

Definition 1.28. (Brouwer Degree) LetU C X := R ^ {N > 1) be a bounded open subset, I G C{U,X), p G X\I{dU). Choose I G C^{Lf,X) such that

sup \\I{u) - I{u)\\ < dist{p,I{dU)) ueu

14 CHAPTER 1. PRELIMINARIES

and define the Brouwer Degree by

deg{I,U,p):=deg{i,U,p),

which is independent of the choice of I.

Proposition 1.29. Let U C X := R^ {N > 1) be a bounded open subset, IeC{U,X),peX\I{dU).

(1)

(2) Let Ui, U2 be two disjoint open subsets ofU,p^ I{U\{Ui U U2)). Then

deg(/, U,p) = deg(/, Ui,p) + deg(/, U2,p).

(3) LetH eC{[^,l] x f 7 , R ^ ) , P G C ( [ 0 , 1 ] , R ^ ) andp{t) ^H{t,dU). Then deg{H{t, •), U^p{t)) is independent of t e [0,1].

(4) (Kronecker Theorem) / /deg(/ , U,p) ^ 0, then there exists a u ^ U such that I{u) = p.

Theorem 1.30. (Borsuk-Ulam Theorem) Let U be an open bounded sym­metric neighborhood ofO in R ^ . Every continuous odd map f : dU -^ R "-*^ has a zero.

As we know, in treating ODE or PDE, the working space is infinite-dimensional. We have to prepare a degree for infinite-dimensional space. Here we introduce a degree for a compact perturbation of the identity. This is the Leray-Schauder degree.

Definition 1.31. Let E be a Banach space, M C E. A mapping I : M ^ E is called compact if I{S) is compact for any bounded subset S of M. Further, if I is continuous, we say that I is completely continuous. In this case, id — I is called a completely continuous field.

Theorem 1.32. Let E be a Banach space, and let M C E be a bounded closed subset. Let I : M ^ E be a continuous mapping. Then I is completely continuous if and only if, for any £ > 0, there exists a finite-dimensional subspace E^ of E and a bounded continuous mapping I^ : M ^ E^ such that

s u p | | /( l^) -In{u)\\ < S. uEM

1.4. TOPOLOGICAL DEGREES 15

Let ^ be a Banach space, and let U C E he di bounded open subset. Let I : U :^ E he completely continuous and f = id — I. If p e E\f{dU), then by Theorem L32, there exists a finite-dimensional subspace E^ of E and a bounded continuous mapping 1^:1)^ E^ such that

s u p | | / ( ^ ) - / ^ H | | < d i s t ( p , / ( a [ / ) ) . ueu

Denote [/ = E^nU; fn{u) = u-In{u), then fn GC(f7n,K),pG En\fn{dUn). Hence, deg(/^, Un^Pn) is well defined.

Definition 1.33. (Leray-Schauder Degree) Let f be the completely con­tinuous field defined as above. Define the Leray-Schauder degree of f at p e E\f{dU) by

^^g{f^U,p) = deg{ fn,Un,p),

which is independent of the choice of En^p^I^.

Proposition 1.34. Let U C E be a bounded open subset of the Banach space E, f = id — I a completely continuous field, and p G E\f{dU). Then

(1)

deg{id,U,p) = ^ J P'

(2) Let [/i, U2 be two disjoint open subsets ofU,p^ f{U\{Ui U U2)), then

deg(/, U,p) = deg(/, Uup) + deg(/, U2,p).

(3) Let H G C([0,1] X U,E) be completely continuous, p G C([0,1],^) and p{t) i H{t, dU) for each t G [0,1]. Then deg{H{t, •), U,p{t)) is indepen­dent oft G [0,1].

(4) (Kronecker Theorem) / /deg( / , U,p) ^ 0, then there exists a u ^ U such that f{u) = p.

Theorem 1.35. (Borsuk-Ulam Theorem) Let U be an open bounded sym­metric neighborhood of 0 in a Banach space E. A completely continuous field f = id — I : U ^ E, where I is odd on dU, p G E\f{dU), then deg(/, U^p) is an odd number.

We also refer the readers to N. G. Lloyd [238], L. Nirenberg [264], M. Nagumo [260, 261], D. Guo [177] and C. Zhong-X. Fan-W. Chen [383] for the details of degree theory.

16 CHAPTER 1. PRELIMINARIES

1.5 An ODE in Banach Space

Let ^ be a Banach space with a norm || • ||. Consider the following Cauchy initial value problem of the ordinary differential equation

(1.14)

d(^ -r ^ / / NX

cr{0,uo) =uo e E,

where V is a potential function. We are interested in the existence of solutions to (1.14), which will play an important role in the following chapters. We assume:

(V) y : ^ ^ ^ is a locally Lipschitz continuous mapping, i.e., for any u ^ E, there exists a ball B{u,r) := {w ^ W : \\w — u\\ < r} with radius r and a constant p > 0 depending on r and u such that

\\V{wi) -V{w2)\\ < p\\wi -W2\\, y wi,W2 e B{u,r).

Moreover,

||y(^)|| < a + 6||^||, yueE,

where a, 6 > 0 are constants.

Theorem 1.36. Assume {V). Then for any u ^ E, the Cauchy prob­lem, (1.14) has a unique solution o-{t,u) defined on the interval [0,+00) of t. Moreover, o-{t,u) depends continuously on the initial data u. Hence, a eC\[0,^oo) X E,E).

To prove Theorem 1.36, we prepare two auxiliary results.

Lemma 1.37. (Gronwah Inequality) If X > 0, (3 > 0 and f e C([0,r],R+) satisfies

(1.15) / ( t ) < A + /3 / f{s)ds, VtG[0 , r ] , Jo

then f{t) < Xe^^ for all t G [0,r].

Proof. By (1.15), we observe that

(e~^' / f{s)ds) < Xe-dt V JQ

Integrating both sides on [0,t], we get the conclusion. D

1.5. AN ODE IN BANACH SPACE 17

Lemma 1.38. (Banach Fixed Point Theorem) Let E be a Banach space, with D C E dosed. Let F : D ^ D satisfy

(1.16) ||Fi^ - Fv\\ < k\\u - v\\ for some k G (0,1) and all u,v e D.

Then there exists a unique u^ such that Fu^ = u^.

Proof. Choose UQ e D and let i^n+i = Fun. Using (1.16) repeatedly, we have

ll'^^n+m+l - Un\\ < (1 - ky^k'^Wui - Uo\\ ^ 0, U^ + 0 0 .

Therefore, {un} is a Cauchy sequence. The conclusion follows from the con­tinuity of F . n

Proof of Theorem 1.36. For any fixed i o ^ E^ by condition (V), we find a ball B{uo,r) := {w e W : \\w — uo\\ < r} with radius r and a constant p > 0 depending on r and i o such that

\\V{wi) -V{w2)\\ < p\\wi -W2\\, y wi,W2 e B{uo,r).

Let A := sup ||y||.

B(uo,r)

Then A < +00. Choose £ > 0 such that sp < l^sA < r. Consider the Banach space

E := C([0,£],F) := {u : [0,^] ^ F is a continuous function}

with the norm ||i^||^ := max^^[o,£] ||'? (OII for each u e E. Let D := {u e E : \\u — uo\\^ < r}. Define a mapping F : E ^ E hj

Fu := 10 + / V{u{s))ds, u e E. Jo

For any u,w ^ D we have

\\Fu-uoh< I \\V{u{s))\\^ds<As<r Jo

and

| F ^ - F ^ | | ^ < max / \\V{u) - V{w)\\^ds < ps\\u - w\\^.

Therefore, F : D ^ D satisfies all conditions of Lemma 1.38. Hence, F has a unique fixed point u^ ^ D, which is a solution of Cauchy problem (1.14).

18 CHAPTER 1. PRELIMINARIES

On the other hand, assume tha t u{t) and v{t) are solutions of the Cauchy problem (1.14) corresponding to initial da ta i o and VQ, respectively. Then

Mt)-v{t)u

<\\UO-VO\\E+ I \\V{u{s))-V{v{s))\\^ds Jo

<ho-vo\\^^p \\u{s)-v{s)\\^ds. Jo

By Lemma 1.37, \\uit)-vit)\\^<\\uo-vohe^K

This proves the continuous dependence on the initial da ta of solution of (1.14). Summing up, (1.14) has a unique solution u{t) on the maximal existence interval [0, K.) which is continuously depending on the initial data. Next, we just show tha t n = +oo. Assume tha t n < +oo. Then

u{t) = UQ -\- V{u{s))ds. Jo

Thus, by ( y ) ,

| |^(t) | | < WuoW^an^b [ \\u{s)\\ds. Jo

Lemma 1.37 implies tha t there is a constant Ci depending on u^^n^a and h such tha t

\\u{t)\\<C,.

It follows tha t \Ht)-u{s)\\<C2\t-s\.

This implies tha t the limit lim u{t) = ui exists. Consider the Cauchy initial

value problem

(1.17)

cr(0,l^l) =Ui e E.

Similarly, it has a unique solution u{t) on a maximal interval [0,/^i) with initial da ta ui = u{n — 0). Let

r u{t), tG[o,/^), v{t) = I

Then v{t) is also a solution of (1.14) with the initial da ta i o on the maximal interval [0, n -\- ni). This produces a contradiction. D

1.6. THE (PS) CONDITIONS 19

1.6 The (PS) Conditions

Many nonlinear problems can be reduced to the form

(1.18) / ' ( ^ ) = 0 ,

where / is a C functional on a Banach space. Equation (1.18) is called the Euler-Lagrange equation of the functional / . The original idea was to find maxima and minima of / , and the critical point theory was devoted to finding extrema of / . The simplest extrema to find are global maxima and minima if / is semibounded. However, we can derive from the semiboundedness the existence of a sequence {un} C E such that

We now introduce definitions of compactness conditions.

Definition 1.39. Any sequence {u^} satisfying

(1.19) sup| / (^^) | <oo , l\ur,)^0, n

is called a Palais-Smale sequence ((PS)-sequence, for short). If any (PS)-sequence of I possesses a convergent subsequence, we say that I satisfies the (PS) condition.

The original idea of the (PS) condition was introduced by R. Palais [269], S. Smale [343] and R. Palais-S. Smale [272]. One of the weak versions of the (PS) condition was proposed in G. Cerami [86].

Definition 1.40. Any sequence {un} satisfying

(1.20) s u p | / ( ^ ^ ) | < OO, (1 + | | ^ n | | ) / ' ( ^ n ) ^ 0, n

is called a Cerami sequence ((C)-sequence, for short). If any (C)-sequence of I possesses a convergent subsequence, we say that I satisfies the (C) condition.

Theorem 1.41. Let E be a Banach space, I G C^(^,R). Assume

I'{u) =Lu^J'{u), ue E,

where L : E ^^ E' is a bounded linear invertible operator and J' maps bounded sets to relatively compact sets in E'. Then any bounded (PS)-sequence or (C)-sequence is relatively compact.

20 CHAPTER 1. PRELIMINARIES

Proof. Let {un} be a bounded (PS)-sequence or (C)-sequence, then I'{un) -^ 0. The conclusion fohows from the relative compactness of J' and Un = L-^r{Un)-L-^J'{Un). •

It is well known, for a given functional / , there may exist critical points of / which are not even local extrema. The existence of this class of critical points was first studied by A. Ambrosotti and P. H. Rabinowitz [19], where a set of sufficient conditions was provided. We also refer the readers to M. Struwe [352]. This abstract critical point theory was based on the (PS) condition. In this book, we will establish a series of theorems without any (PS) type assumption.

1.7 Weak Solutions

In practice, one of the main research projects using critical point theory is the existence of solutions to elliptic equations. For example, consider

(1.21) -/\u = f{x,u), xen.

The corresponding functional is defined by

I{u) = ^\\\/uf- J F{x,u)dx,

where F{x,u) = / f{x,s)ds. Roughly speaking, if / is in C on a suitable Jo

space and I\u) = 0 (a critical point), then

(1.22) / Vu'Vwdx- / f{x,u)wdx = 0, \/w e E. JQ JQ

The critical point u satisfying (1.22) is called a weak solution of (1.21) and ob­viously u is not necessarily a classical solution. In general, more assumptions on the smoothness of dfl and of / are needed if we want the weak solution to be a classical solution. It is not an easy task.

In this section, we just give a simple example to show how the regularity theory of elliptic equations can be applied to obtain a classical solution from a weak solution.

Definition 1.42. Consider a bounded domain ft C R ^ with boundary dft. Let k be a nonnegative integer and o G [0,1]. O is of class C^'^ if at each point xo G dft there is a ball B = B{XQ) and a one-to-one mapping Lp from B onto a subset D C R ^ such that