13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

13.2 Convexity and lower semi-continuity . . . . . . . . . . . . . . . . . . 149

13.3 Existence of saddle points . . . . . . . . . . . . . . . . . . . . . . . 152

13.4 Criteria for convexity . . . . . . . . . . . . . . . . . . . . . . . . . . 155

13.5 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

13.6 The theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

13.7 The proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

13.8 Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

14 Type (II) Regions 163

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

14.2 The asymptotic equation . . . . . . . . . . . . . . . . . . . . . . . . 166

14.3 Local estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

14.4 The solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

14.5 Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

15 Weak Sandwich Pairs 177

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

15.2 Weak sandwich pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 178

15.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

15.4 Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

16 Multiple Solutions 191

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

16.2 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

16.3 Statement of the theorems . . . . . . . . . . . . . . . . . . . . . . . 192

16.4 Some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

16.5 Local linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

16.6 The proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

16.7 Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

x Contents

17 Second-Order Periodic Systems 213

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

17.2 Proofs of the theorems . . . . . . . . . . . . . . . . . . . . . . . . . 215

17.3 Nonconstant solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 221

17.4 Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Bibliography 229

Index 241

Preface

Many problems in science involve the solving of differential equations or systems of

differential equations. Moreover, many of these equations and systems come from

variational considerations involving mappings (called functionals) into the real number

system. As a simple example, consider the problem of ¬nding a solution of

u (x) = f (x, u(x)), x ∈ I = [a, b],

(1)

under the conditions

u(a) = u(b) = 0.

(2)

Assume that the function f (x, t) is continuous in I — R. The corresponding functional

is

b

[(u )2 + 2F(x, u)] d x,

G(u) =

(3)

a

where

t

f (x, s) ds.

F(x, t) :=

0

It is easy to show that u(x) is a solution of the problem (1), (2) if and only if it satis¬es

G (u) = 0.

(4)

Thus, in such cases, solutions of the equations or systems are critical points of the

corresponding functional. As a result, anyone who is interested in obtaining solutions

of the equations or systems is also interested in obtaining critical points of the corre-

sponding functionals. The latter problem is the subject of this book.

The classical way of obtaining critical points was to search for maxima or minima.

This is possible if the functional is bounded from above or below. However, when this

is not the case, there is no organized way of ¬nding critical points.

Linking theory is an attempt to “level the playing ¬eld,” i.e., to ¬nd a substitute for

semiboundedness. It ¬nds a pair of subsets A, B of the underlying space that allow

the functional to have the same advantages as semibounded functionals if the subsets

separate the functional. They separate a functional G if

sup G < inf G.

(5)

B

A

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 publica-

tion 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 satis¬es

sup G < inf sup G,

(6)

K ∈K K

A

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 de¬ne minimax systems and show what they can accom-

plish. 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 Chap-

ter 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

¬nite-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 in¬nite-

dimensional, 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 suf¬cient.

In Chapter 7 we ask the question: Is there anything that can be done if one cannot

¬nd 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:

’∞ < inf G ¤ sup G < ∞.

(7)

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

’ u = f (x, u), x ∈ ; u = 0 on ‚ ,

(8)