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

0 < μF(x, t) ¤ t f (x, t), |t| ≥ r.

(9)

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)/t 2 ’ ∞ as t ’ ∞

or

F(x, t)/t 2 ’ ∞ as t ’ ’∞.

We shall show that under this condition the boundary value problem

’ u = β f (x, u), x ∈ ; u = 0 on ‚ ,

(10)

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 ¬nite 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 Fuˇ´k spectrum. They deal with the situation

c±

concerning the semilinear problem

Au = f (x, u), u ∈ D(A),

(11)

in which

f (x, t)/t ’ a a.e. as t ’ ’∞

’ b a.e. as t ’ +∞,

and

Au = bu + ’ au ’ , u ± = max{±u, 0},

(12)

has a nontrivial solution. We call the set of those (a, b) ∈ R2 for which (12) has

nontrivial solutions the Fuˇ ´k spectrum of A. When (a, b) ∈ , it is more dif¬cult to

c±

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 dif¬culty than others.

Chapter 14 deals with such points.

Chapters 12 and 13 are concerned with the multidimensional wave equation

u ≡ ut t ’ u = p(x, t, u),

(13)

u(x, t) = 0, t ∈ R, x ∈ ‚ .

(14)

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 veri¬ed as sandwich pairs unless at least

one of the sets is contained in a ¬nite-dimensional manifold. We show in Chapter 15

that they can both be in¬nite-dimensional if we add an assumption on G that is satis¬ed

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

’x(t) = ∇x V (t, x(t)),

¨

(15)

where

x(t) = (x 1 (t), . . . , x n (t)).

(16)

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

G (u) = 0,

(1.1)

where G(u) is a C 1 -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 de¬nitions 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 ¬nding 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

G(u) = ∇u ’

2

F(x, u(x)) d x

2

on an appropriate space, where

t

F(x, t) = f (x, s) ds,

(1.2)

0

and the norm is that of L 2 . 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,

© Birkh¤user Boston, a part of Springer Science+Business Media, LLC 2009

2 1. Critical Points of Functionals

more dif¬cult to solve the Euler“Lagrange equations than it was to ¬nd minima or

maxima of functionals. Consequently, the approach was abandoned for many years.

Eventually, when nonlinear partial differential equations and systems arose in applica-

tions 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 ¬nd critical points of the corresponding functionals. The problem is that there is no

uniform way of ¬nding them.

1.2 Extrema

The usual approach to ¬nding critical points was to look for maxima or minima. Global

extrema are the easiest to ¬nd, but they can exist only if the functional is semibounded.

For instance, if the continuously differential functional G is bounded from below, then