2 0. MORSE THEORY AND VARIATIONAL PROBLEMS

0.1. Compactness Conditions

It is usually necessary to assume some sort of a “compactness condition”

when seeking critical points of a functional. The following condition was

originally introduced by Palais and Smale [91]: Φ satisfies the Palais-Smale

compactness condition at the level c, or pPSqc for short, if every sequence

puj q Ă W such that

Φpuj q Ñ c,

Φ1puj

q Ñ 0,

called a pPSqc sequence, has a convergent subsequence; Φ satisfies pPSq if it

satisfies pPSqc for every c P R, or equivalently, if every sequence such that

Φpuj q is bounded and Φ1puj q Ñ 0, called a pPSq sequence, has a convergent

subsequence.

The following weaker version was introduced by Cerami [25]: Φ satisfies

the Cerami condition at the level c, or pCqc for short, if every sequence such

that

Φpuj q Ñ c,

`

1 ` }uj }

˘

Φ1puj

q Ñ 0,

called a pCqc sequence, has a convergent subsequence; Φ satisfies pCq if it

satisfies pCqc for every c, or equivalently, if every sequence such that Φpuj q is

bounded and

`

1 ` }uj }

˘

Φ1puj

q Ñ 0, called a pCq sequence, has a convergent

subsequence. This condition is weaker since a pCqc (resp. pCq) sequence is

clearly a pPSqc (resp. pPSq) sequence also.

The limit of a pPSqc (resp. pPSq) sequence is in

Kc

(resp. K) since Φ

and

Φ1

are continuous. Since any sequence in

Kc

is a pCqc sequence, it

follows that

Kc

is a compact set when pCqc holds.

0.2. Deformation Lemmas

An essential tool for locating critical points is the deformation lemmas,

which allow to lower sublevel sets of a functional, away from its critical set.

The main ingredient in their proofs is a suitable negative pseudo-gradient

flow, a notion due to Palais [93]: a pseudo-gradient vector field for Φ on

Ă

W

is a locally Lipschitz continuous mapping V :

Ă

W Ñ W satisfying

}V puq} ď

›

›Φ1puq›˚

›

, 2

Φ1puq,V

puq ě

`›

›Φ1puq›˚

› ˘2

@u P

Ă

W.

Such a vector field exists, and may be chosen to be odd when Φ is even.

The first deformation lemma provides a local deformation near a (pos-

sibly critical) level set of a functional.

Lemma 0.1 (First Deformation Lemma). If c P R, C is a bounded set

containing

Kc,

δ, k ą 0, and Φ satisfies pCqc, then there are an ε0 ą 0 and,

for each ε P p0, ε0q, a map η P Cpr0, 1s ˆ W, W q satisfying

piq ηp0, ¨q “ id

W

,

piiq ηpt, ¨q is a homeomorphism of W for all t P r0, 1s,

piiiq ηpt, ¨q is the identity outside A “ Φc´2εzNδ{3pCq

c`2ε

for all t P r0, 1s,

pivq }ηpt, uq ´ u} ď

`

1 ` }u}

˘

δ{k @pt, uq P r0, 1s ˆ W ,

pvq Φpηp¨,uqq is nonincreasing for all u P W ,