CHAPTER 0 Morse Theory and Variational Problems In this preliminary chapter we give a brief survey of Morse theoretic methods used in variational problems. General references are Milnor [84], Mawhin and Willem [81], Chang [28], and Benci [19]. We consider a real-valued function Φ defined on a real Banach space pW, }¨}q. We say that Φ is Fr´ echet differentiable at u P W if there is an element Φ1puq of the dual space pW ˚ , }¨}˚q, called the Fr´ echet derivative of Φ at u, such that Φpu ` vq Φpuq ` Φ1puq,v ` op}v}q as v Ñ 0 in W, where ¨, ¨ is the duality pairing. The functional Φ is continuously Fr´echet differentiable on W , or belongs to the class C1pW, Rq, if Φ1 is defined every- where and the map W Ñ W ˚ , u Þ Ñ Φ1puq is continuous. We assume that Φ P C1pW, Rq for the rest of the chapter. We say that u is a critical point of Φ if Φ1puq 0. A real number c P ΦpW q is a critical value of Φ if there is a critical point u with Φpuq c, otherwise it is a regular value. We use the notations Φa u P W : Φpuq ě a ( , Φb u P W : Φpuq ď b ( , Φa b Φa X Φb, K u P W : Φ1puq 0 ( , Ă W zK, Kb a K X Φb, a Kc Kc c for the various superlevel, sublevel, critical, and regular sets of Φ. We begin by recalling the compactness condition of Palais and Smale and its weaker variant given by Cerami in Section 0.1. Then we state the first and second deformation lemmas under the Cerami’s condition in Section 0.2. In Section 0.3 we define the critical groups of an isolated critical point and summarize the basic results of Morse theory. These include the Morse inequalities, Morse lemma and its generalization splitting lemma, shifting theorem of Gromoll and Meyer, and the handle body theorem. Next we dis- cuss the minimax principle in Section 0.4. Section 0.5 contains a discussion of homotopical linking, pairs of critical points with nontrivial critical groups produced by homological linking, and nonstandard geometries without a fi- nite dimensional closed loop. We recall the notion of local linking and an alternative for a critical point produced by a local linking in Section 0.6. We conclude with a result on nontrivial critical groups associated with the p-Laplacian in Section 0.7. 1
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