Schechter Minimax Systems and Critical Point Theory (Springer, 2009)

xii

Preface

This is the theme of the book [122], which records much of the work of researchers on this approach up to that time.

There are several methods of obtaining linking sets (some will be outlined later in Chapters 3 and 6). The purpose of the present volume is to unify some of these approaches and to study results and applications that were obtained since the publication of [122].

The underlying theme is to consider minimax systems depending on a set A. These are collections K of subsets such that if the functional G satisfies

(6)	sup G <	inf sup G,
	A	K K K

then it has the same advantages as a semibounded functional. We show that the main approaches to linking can be combined by using minimax systems.

In Chapters 1 and 2, we define minimax systems and show what they can accomplish. We consider several variations and generalizations that are useful in applications.

In Chapter 3 we describe some methods used by researchers to obtain critical points of functionals, and we show that these results are contained in the theorems of Chapter 2. Various geometries are considered. In particular, the sandwich theorem of [143], [109], and [108] is generalized. This considers the following situation when N is a finite-dimensional subspace of a Hilbert space E and M = N . If G is a functional on E such that G is bounded from below on M and bounded from above on N, then G has the advantages of a semibounded functional. If both subspaces are infinitedimensional, it may be necessary to impose additional assumptions on the functional G. This is discussed in Chapter 15.

In Chapter 4 we prove some theorems concerning differential equations in abstract spaces. These results are needed in proving the theorems of Chapter 2.

In Chapter 5 we give the proofs of the theorems of Chapter 2 using the results of Chapter 4.

In Chapter 6 we search for linking subsets. We believe in principle that we have found most, if not all, of them, at least in an abstract way. We produce two criteria for subsets to satisfy, one slightly stronger than the other. The weaker criterion is necessary for linking, while the stronger is sufficient.

In Chapter 7 we ask the question: Is there anything that can be done if one cannot find linking subsets that separate the functional? A surprising answer is yes. We show that there are subsets A, B that produce the same effect when they do not separate the functional, i.e., when they satisfy just the opposite:

(7) −∞ < inf G ≤ sup G < ∞.

B A

We call such sets a sandwich pair. The reason is that the subspaces described in the sandwich theorem of Chapter 3 form a sandwich pair. We describe criteria for obtaining sandwich pairs.

In Chapter 8 we describe applications of the theories presented in Chapters 1–7. Some are more involved than those usually found in the literature. Other applications are given in Chapters 9–17.

Preface	xiii
In Chapter 9 we describe superlinear problems of the form
(8)	− u = f (x, u), x ; u = 0 on ∂ ,

where Rn is a bounded domain whose boundary is a smooth manifold and f (x, t)

is a continuous function on ¯ × R The problem is called superlinear if f x t does

. ( , ) not satisfy an inequality of the form

| f (x, t)| ≤ C(|t| + 1), x , t R.

Almost all researchers who studied the superlinear problem make the assumption that there are constants μ > 2, r ≥ 0 such that

(9) 0 < μF(x, t) ≤ t f (x, t), |t| ≥ r.

This is a very convenient assumption, but it excludes a large number of superlinear problems. An assumption that is much weaker is: Either

F(x, t)/t2 → ∞ as t → ∞

or

F(x, t)/t2 → ∞ as t → −∞.

We shall show that under this condition the boundary value problem

(10)	− u = β f (x, u), x ; u = 0 on ∂ ,

has a nontrivial solution for almost every positive β. We require a stronger hypothesis to show that the statement is true for any particular β.

A fact of life which plagues researchers is that we can verify that subsets link only if at least one of them is contained in a finite dimensional manifold. In Chapter 10 we deal with this problem by adding an assumption to the functional. We try to make this assumption as weak as possible. Applications are given.

Chapters 11 and 14 deal with the Fuˇc´ık spectrum. They deal with the situation concerning the semilinear problem

(11)	Au = f (x, u), u D( A),
in which
	f (x, t)/t → a a.e. as t → −∞
	→ b a.e. as t → +∞,
and
(12)	Au = bu+ − au−, u± = max{±u, 0},

has a nontrivial solution. We call the set of those (a, b) R2 for which (12) has nontrivial solutions the Fuˇc´ık spectrum of A. When (a, b) , it is more difficult to

xiv

Preface

solve (11) than when (a, b) / . Chapter 11 deals with solving (11) when (a, b). Moreover, not all (a, b) / are alike; some cause more difficulty than others. Chapter 14 deals with such points.

Chapters 12 and 13 are concerned with the multidimensional wave equation

(13)	u ≡ ut t − u = p(x, t, u),
(14)	u(x, t) = 0, t R, x ∂ .

In Chapter 12 we study rotationally invariant solutions. In Chapter 13 we study the more general situation.

In Chapter 15 we discuss systems of equations for which the method of sandwich pairs would be ideal. However, sandwich pairs are plagued by the same problem as linking subsets, namely, that they cannot be verified as sandwich pairs unless at least one of the sets is contained in a finite-dimensional manifold. We show in Chapter 15 that they can both be infinite-dimensional if we add an assumption on G that is satisfied in most applications.

In Chapter 16 we show that in many situations, nonlinear boundary-value problems have multiple solutions.

Chapter 17 deals with second-order periodic systems of the form

(15)	−x¨ (t) = x V (t, x(t)),
where
(16)	x(t) = (x1(t), . . . , xn (t)).

We give several sets of conditions that imply the existence of solutions and conditions which imply the existence of nonconstant solutions.

Martin Schechter

Irvine, California

TVSLB O

Chapter 1

Critical Points of Functionals

1.1 Introduction

Many problems arising in science and engineering call for the solving of the Euler–Lagrange equations of functionals, i.e., equations equivalent to

(1.1)

G (u) = 0,

where G(u) is a C1-functional (usually representing the energy) arising from the given data. By this we mean that functions are solutions of the Euler–Lagrange equations of G iff they satisfy (1.1). (For various definitions of the derivative G (u) of G, cf., e.g., [127].) Solutions of (1.1) are called critical points of G. Thus, solving the Euler–Lagrange equations is tantamount to finding critical points of the corresponding functional.

As an illustration, the equation

	− u(x) = f (x, u(x))
is the Euler–Lagrange equation of the functional
	1		u 2 − F(x, u(x)) dx
	G(u) =
		2
on an appropriate space, where
	F(x, t) =				t
(1.2)				0	f (x, s) ds,

and the norm is that of L2. The variational approach to solving differential equations and systems has its roots in the calculus of variations. The original problem was to minimize or maximize a given functional. The approach was to obtain the Euler– Lagrange equations of the functional, solve them, and show that the solutions provided the required minimum or maximum. This worked well for one-dimensional problems. However, when it came to higher dimensions, it was recognized quite early that it was

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_1, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

2	1. Critical Points of Functionals

more difficult to solve the Euler–Lagrange equations than it was to find minima or maxima of functionals. Consequently, the approach was abandoned for many years. Eventually, when nonlinear partial differential equations and systems arose in applications and people were searching for solutions, they began to check if the equations and systems were the Euler–Lagrange equations of functionals. If so, a natural approach is to find critical points of the corresponding functionals. The problem is that there is no uniform way of finding them.

1.2 Extrema

The usual approach to finding critical points was to look for maxima or minima. Global extrema are the easiest to find, but they can exist only if the functional is semibounded. For instance, if the continuously differential functional G is bounded from below, then we can find a minimizing sequence {uk } such that

(1.3)

G(uk ) → a = inf G > −∞.

If this series converges or has a convergent subsequence, we have a minimum. However, we have no guarantee that a subsequence will converge. Moreover, a minimizing sequence provides very little structure that can help one find a convergent subsequence. Indeed, one can find simple examples of functionals that are bounded below and have no minimum. For instance, the function

y(x) = ex

has no minimum on R even though it is bounded from below.

1.3 Palais–Smale sequences

On the other hand, if the functional G is bounded from below, it can be shown that there is a sequence satisfying

(1.4)

G(uk ) → a, G (uk ) → 0.

(Actually, one can do better; see below.) If the sequence has a convergent subsequence, this will produce a minimum. Such a sequence is called a Palais–Smale (PS) sequence. The advantage of having a PS sequence instead of a minimizing sequence is that the additional information

G (uk ) → 0

helps one show that a PS sequence has a convergent subsequence more readily than one can show that a minimizing sequence has a convergent subsequence.

1.4. Cerami sequences

3

1.4 Cerami sequences

Actually, it can be shown that a C1-functional G that is bounded from below has a Cerami sequence, a sequence satisfying

(1.5)

G(uk ) → a, (1 + uk )G (uk ) → 0

(cf. Theorem 3.21 and its corollary). A Cerami sequence says more. There are examples for which one can show that a Cerami sequence has a convergent subsequence, while one cannot do the same for the corresponding PS sequence.

1.5 Linking sets

When the functional is not semibounded, there is no clear way of obtaining critical points. In general, one would like to determine when a functional has PS or Cerami sequences. That is, one would like to find situations that give one the same advantages that one has in the case of semibounded functionals.

An idea that has been very successful is to find appropriate sets that separate the functional. By this we mean the following:

Definition 1.1. Two sets A, B separate the functional G(u) if

(1.6)	a0 := sup G < b0 :=	inf G	.
		B
	A

We would like to find sets A and B such that (1.6) will imply

(1.7)

u : G(u) ≥ b0, G (u) = 0.

This is too much to expect, since even semiboundedness alone does not imply the existence of a critical point. Consequently, we weaken our requirements and look for sets A, B such that (1.6) implies the existence of a PS sequence (1.4) with a ≥ b0. This leads to

Definition 1.2. We shall say that the set A links the set B if (1.6) implies (1.4) with a ≥ b0 for every C1-functional G(u).

Of course, (1.4) is a far cry from (1.7), but if, e.g., the sequence (1.4) has a convergent subsequence, then (1.4) implies (1.7). Whether or not (1.4) implies (1.7) is a property of the functional G(u). We state this as

Definition 1.3. We say that G(u) satisfies the PS condition if (1.4) always implies (1.7).

The usual way of verifying this is to show that every sequence satisfying (1.4) has a convergent subsequence (there are other ways).

4	1. Critical Points of Functionals

All of this leads to

Theorem 1.4. If G satisfies the PS condition and is separated by a pair of linking sets, then it has a critical point satisfying (1.7).

This theorem cannot be applied until one identifies linking sets and functionals that satisfy the PS condition. Fortunately, they exist. We shall describe many of the known linking sets later in the book.

Among other things, we shall discuss when

(1.8)	a0 := sup G ≤ b0 :=	inf G
		B
	A

implies the existence of a PS sequence or a Cerami sequence.

If there do not exist linking sets that separate a functional, all is not lost. There exist sets that produce PS or Cerami sequences when they do not separate a functional. That is, we shall find sets such that

(1.9)	−∞ <	b	0 :=	B	,	a	0	=	A	< ∞
				inf G				:	sup G

implies the existence of a PS sequence or a Cerami sequence.

1.6 Previous definitions of linking

In the earlier versions of linking, A is a compact set and is the “boundary” of a manifold S. They say that A links a set B if

A ∩ B = φ

and every continuous map ϕ from S to E that equals the identity on A must satisfy

ϕ(S) ∩ B = φ.

The underlying theorem is

Theorem 1.5. If the set A links the set B and (1.6) holds, then there is a Palais–Smale sequence satisfying (1.4) with a ≥ b0.

There are distinct disadvantages of this definition. First, it requires A to be compact. Second, it requires A to be the “boundary” of a manifold S. Third, linking depends on the manifold S. Finally, there is no possibility of symmetry in infinite-dimensional spaces, i.e., if A links B, then B cannot link A according to this definition.

Several methods have been used to circumvent these shortcomings and extend the definition of linking to cover cases when the old definition does not apply. Some of them are described in Chapter 3. The purpose of this volume is to present a uniform approach that includes most, if not all, theories of linking. It employs minimax systems and is presented in Chapter 2.

The outline of this book is as follows. In Chapter 5 we give the proofs of the theorems of Chapter 2. We use methods of solving differential equations in Banach

1.7. Notes and remarks

5

spaces described in Chapter 4. In Chapter 6 we show that our uniform method is almost identical to the general Definition 1.2 given above. In Chapter 7 we consider the case when there are no linking sets that separate the functional. We find pairs of sets (called sandwich pairs) that produce PS sequences (1.4) when (1.6) fails. In Chapters 8, 9, 11, 14, 16, and 17 we give some applications to partial differential equations of the methods presented in Chapters 1–7. In Chapters 10, 12, 13, and 15 we are concerned with the situation when A and B are both infinite-dimensional.

1.7 Notes and remarks

For an excellent review of critical point theory and linking, we refer to [96]. Previous definitions of linking theory are described in [8]. Other definitions of linking will be discussed in Chapters 2 and 3.

Chapter 2

Minimax Systems

2.1 Introduction

We have described how the variational approach to solving nonlinear problems eventually leads to the search for critical points of related functionals. In the case of semibounded functionals, one can look for extrema. Otherwise, one is forced to use other methods. As we mentioned, linking provides a useful tool. There are several approaches to linking. In this book we unify these approaches, providing one theory that works for all of them.

The idea is as follows. For each A E, one wishes to find a collection K of sets

Kwith the following properties.

1.If G C1(E, R) satisfies

(2.1)	a0 := sup G < a := Kinf sup G,
	A	K K

then there is a sequence {uk } E such that (1.4) holds. 2. If

(2.2)	a0 ≤ sup G, K K,
	K \ A

then there is a sequence {uk } E such that (1.4) holds. 3. If A, B E satisfy

(2.3)	A ∩ B = φ,			B ∩ K = φ,		K K,
and
(2.4)	a	0	:= sup G < b0 :=		inf G		,
					B
				A

then there is a sequence {uk } E such that (1.4) holds.

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_2, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

8	2. Minimax Systems
4. If
(2.5)	gK = {v K \ A : G(v) ≥ a0} = φ, K K,

then there is a sequence {uk } E such that (1.4) holds.

In the next section we determine collections K that have these properties. Moreover, we shall show that all of these collections produce not only PS sequences (1.4), but also Cerami sequences (1.5) as well.

Proofs of the theorems of this chapter will be given in Chapter 5.

2.2 Definitions and theorems

We begin by studying C1-functionals on a Banach space E.

Definition 2.1. We shall say that a map ϕ : E → E is of class if it is a homeomorphism onto E, and both ϕ, ϕ−1 are bounded on bounded sets.

Definition 2.2. For A E, we define

( A) = {ϕ : ϕ(u) = u, u A}.

Definition 2.3. For a nonempty set A E, we define a nonempty collection K = K( A) of subsets K E to be a minimax system for A if it has the following property:

ϕ(K ) K, ϕ ( A), K K.

Every nonempty set has a minimax system. We have

Theorem 2.4. Let K be a minimax system for a nonempty subset A of E, and let G(u) be a C1-functional on E. Define

(2.6)	a := Kinf sup G,
	K K
and assume that a is finite and satisfies
(2.7)	a > a0 := sup G.
	A

Let ψ(t) be a positive, nonincreasing, locally Lipschitz continuous function on [0, ∞) such that

(2.8)			∞	ψ(r ) dr = ∞.
			0
Then there is a sequence {uk } E such that
(2.9)	G(uk )	→	a,	G (uk )/ψ( uk	)	0.
					→
Definition 2.5. We shall call σ (t) C([0, T ] × E, E), T						> 0, a flow if σ (0) = I

and for each t [0, T ], σ (t) is a homeomorphism of E onto itself.