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.
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