4

AN OVERVIEW

Let B

r

denot e the open ball in E o f radius r abou t 0 and let dB

r

denot e its

boundary. No w the Mountain Pass Theorem can be stated.

THEOREM. Let E be a real Banach space and I € C

X(E,R).

Suppose I

satisfies {PS), 1(0) = 0,

(Ji) there exist constants p,a 0 such that I\dBp OL, and

(h) there is aneeE\dBp such that I(e) 0.

Then I possesses a critical value ca which can be characterized as

c = in f ma x I(u).

gerueglOA]

where

r = {g G c([o, i],E)\g(o) = 0,9(1) = e.

This result i s due to Ambrosetti and Rabinowit z [AR] . On a heuristic level,

the theorem says if a pair of points in the graph of / ar e separated by a mountain

ränge, there must be a mountain pass containing a critical point between them.

Although the Statement of the theorem does not require it, in applications it is

generally the case that I ha s a local minimum at 0.

A seeond geometrica l exampl e of a minima x resul t i s th e followin g Saddl e

Point Theorem [R4]:

THEOREM. Let E be a real Banach space such that E = V © X, where V is

finite dimensionaL Suppose I £ C

1(£',R),

satisfies (PS), and

(Is) there exists a bounded neighborhood, D, o/ 0 in V and a constant a such

that I\dD ^ « i ö^ d

(I4) there is a constant ß a such that I\x ß-

Then I has a critical value c ß. Moreover c can be characterized as

c = in f max/fti) ,

where

r = {S = h(D)\h € C(D, E) and h = idon dD).

Here heuristically c is the minimax of I ove r all surfaces modelled on D an d

which share the same boundary. Unlik e the Mountain Pas s Theorem, in appli-

cations of the Saddle Point Theorem generally no critical points of / ar e known

initially. Not e that (J3 ) and (I4) are satisfied i f i" is convex on X, concav e on V,

and appropriately coercive. Indeed the Saddle Point Theorem was motivated by

earlier results of that natur e due to Ahmad, Lazer, and Paul [ALP] and Castro

and Lazer [CL].

Both of the above theorems, generalizations, and applications will be treated

in Chapters 2-6 . I n particular i n a somewhat mor e restrictive setting both the

Mountain Pass Theorem and Saddle Point Theorem can be interpreted as special

cases of a more general critical point theorem which is proved in Chapter 5.

Much of the remainder of these lectures will be devoted to the study of vari-

ational problems in which symmetries play a role. T o be more precise, suppose