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 is a locally Lipschitz continuous mapping V : Ă Ñ W satisfying }V puq} ď › ›Φ1puq›˚ › , 2 Φ1puq,V puq ě `› ›Φ1puq›˚ › ˘ 2 @u P Ă 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`2εzN c´2ε δ{3 pCq 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 ,

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