Let B
denot e the open ball in E o f radius r abou t 0 and let dB
denot e its
boundary. No w the Mountain Pass Theorem can be stated.
THEOREM. Let E be a real Banach space and I C
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).
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
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) ,
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
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