6 0. MORSE THEORY AND VARIATIONAL PROBLEMS

Splitting lemma when A is a compact perturbation of the identity was

proved by Gromoll and Meyer [54] for Φ P

C3

and by Hofer [56] in the

C2

case. Mawhin and Willem [80, 81] extended it to the case where A is a

Fredholm operator of index zero. The general version given here is due to

Chang [28].

A consequence of the splitting lemma is

Theorem 0.10 (Shifting Theorem). We have

CqpΦ, 0q “ Cq´mp0qpΦ|N , 0q @q

where N “ ξpB X N q is the degenerate submanifold of Φ at 0.

Shifting theorem is due to Gromoll and Meyer [54]; see also Mawhin and

Willem [81] and Chang [28].

Since dim N “ m˚p0q ´ mp0q, shifting theorem gives us the following

Morse index estimates when there is a nontrivial critical group.

Corollary 0.11. If CkpΦ, 0q ‰ 0, then

mp0q ď k ď

m˚p0q.

It also enables us to compute the critical groups of a mountain pass

point of nullity at most one.

Theorem 0.12. If u is a mountain pass point of Φ and dim ker

Φ2puq

ď 1,

then

CqpΦ,uq “ δq1 G.

This result is due to Ambrosetti [5, 6] in the nondegenerate case and to

Hofer [56] in the general case.

Shifting theorem also implies that all critical groups of a critical point

with infinite Morse index are trivial, so the above theory is not suitable

for studying strongly indefinite functionals. An infinite dimensional Morse

theory particularly well suited to deal with such functionals was developed

by Szulkin [128]; see also Kryszewski and Szulkin [61].

The following important perturbation result is due to Marino and Prodi

[78]; see also Solimini [125].

Theorem 0.13. If some critical value of Φ has only a finite number of

critical points ui, i “ 1,...,k and

Φ2puiq

are Fredholm operators, then for

any suﬃciently small ε ą 0 there is a

C2-functional

Φε on H such that

piq }Φε ´ Φ}C2pHq ď ε,

piiq Φε “ Φ in Hz

Ťk

i“1

Bεpuj q,

piiiq Φε has only nondegenerate critical points in Bεpuj q and their Morse

indices are in rmpuiq,

m˚puiqs,

pivq Φ satisfies pPSq ùñ Φε satisfies pPSq.