As a secon d exampl e o f a n indefinit e functional , conside r th e Hamiltonia n
System of ordinary differential equation s
(1.6) ^ = -H g(p,q), %=H p(p,q),
where H: R
* R i s smooth, an d p and q are n-tuples. W e are interested i n
periodic Solutions of (1.6). Taking the period to be 2it and choosing E t o be an
appropriate space of 2?r periodic functions, Solution s of (1.6) are critical point s
/•27 T
(1.7) I(p, q) = / \p(t) q(t) - H(p(th q(t))] dt.
(This will be made precise in Chapter 6. ) T o see the indefinite natur e of (1.7),
suppose n = 1. Taking pk(t) = sin kt and&(£) = - co s kt shows I(pk, qk) = kit+
bounded term—• ±oo as k —• ±oo. Thu s I i s not bounded from above or below.
Despite this , a s w e shal l se e later , minima x method s ca n b e applie d t o th e
functional (1.7) to obtain periodic Solutions of (1.6).
There ar e a t leas t tw o set s o f method s tha t hav e bee n develope d t o find
critical point s o f functionals: (i ) Mors e theory an d it s generalization s an d (ii )
minimax theory . Fo r materia l o n "classical " Mors e theory , se e e.g. [Mi , S2 ,
Ch2]. Generalize d Morse theories and the so-called Conley index can be found
in th e CBM S monograph o f Conle y [CC ] (se e als o [Sm]) . Ou r lecture s wil l
focus o n minima x theory . Thi s subjec t originate d i n wor k o f Ljusterni k an d
Schnirelman [LLS ] although it certainly had antecedents (se e e.g. [Bi]).
What are minimax methods? Thes e are methods that characteriz e a critical
value c of a functional I a s a minimax over a suitable dass of sets 5 :
(1.8) c = in f maxJ(u) .
There is no recipe for choosing 5. I n any given Situation the choice must reflect
some qualitative change in the topological nature of the level sets of I, i.e . the
fo r s near c. Thu s obtaining and characterizing a critical value c as
in (1.8) is something of an ad hoc process.
The Mountain Pass Theorem is the first minimax result that we will study. It s
statement involves a useful technical assumption—the Palais-Smale condition—
that occur s repeatedl y i n critica l poin t theory . Suppos e E i s a rea l Banac h
space. Let
(15, R) denote the set of functionals tha t are Frechet difFerentiabl e
and whose Frechet derivatives are continuous on E. Fo r I C
R), we say I
satisfies the Palais-Smale condition (henceforth denoted by (PS)) if any sequence
(um) C E fo r whic h I{u m) i s bounded an d I'(u m) - ^ O a s m - ^ o o possesse s
a convergen t subsequence . Th e (PS ) conditio n i s a convenien t wa y t o buil d
some "compactness " int o th e functiona l L Indee d observ e tha t (PS ) implie s
that K
= {u G E\I(u) = c and I
= 0} , i.e. the set of critical points having
critical value c, is compact fo r an y cGR. W e will see many examples later of
when (PS) is satisfied.
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