CHAPTER 0

Morse Theory and Variational Problems

In this preliminary chapter we give a brief survey of Morse theoretic

methods used in variational problems. General references are Milnor [84],

Mawhin and Willem [81], Chang [28], and Benci [19].

We consider a real-valued function Φ defined on a real Banach space

pW, }¨}q. We say that Φ is Fr´ echet differentiable at u P W if there is an

element

Φ1puq

of the dual space pW

˚,

}¨}˚q,

called the Fr´ echet derivative of

Φ at u, such that

Φpu ` vq “ Φpuq `

Φ1puq,v

` op}v}q as v Ñ 0 in W,

where ¨, ¨ is the duality pairing. The functional Φ is continuously Fr´echet

differentiable on W , or belongs to the class

C1pW,

Rq, if

Φ1

is defined every-

where and the map W Ñ W

˚,

u Þ Ñ

Φ1puq

is continuous. We assume that

Φ P

C1pW,

Rq for the rest of the chapter. We say that u is a critical point

of Φ if

Φ1puq

“ 0. A real number c P ΦpW q is a critical value of Φ if there is

a critical point u with Φpuq “ c, otherwise it is a regular value. We use the

notations

Φa “ u P W : Φpuq ě a

(

,

Φb

“ u P W : Φpuq ď b

(

, Φa

b

“ Φa X

Φb,

K “ u P W :

Φ1puq

“ 0

(

,

Ă

W “ W zK, Ka

b

“ K X Φa,

b Kc

“

Kcc

for the various superlevel, sublevel, critical, and regular sets of Φ.

We begin by recalling the compactness condition of Palais and Smale and

its weaker variant given by Cerami in Section 0.1. Then we state the first

and second deformation lemmas under the Cerami’s condition in Section

0.2. In Section 0.3 we define the critical groups of an isolated critical point

and summarize the basic results of Morse theory. These include the Morse

inequalities, Morse lemma and its generalization splitting lemma, shifting

theorem of Gromoll and Meyer, and the handle body theorem. Next we dis-

cuss the minimax principle in Section 0.4. Section 0.5 contains a discussion

of homotopical linking, pairs of critical points with nontrivial critical groups

produced by homological linking, and nonstandard geometries without a fi-

nite dimensional closed loop. We recall the notion of local linking and an

alternative for a critical point produced by a local linking in Section 0.6.

We conclude with a result on nontrivial critical groups associated with the

p-Laplacian in Section 0.7.

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http://dx.doi.org/10.1090/surv/161/01