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 Ť k 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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