Abstract
In this thesis, we present systematically the Palais-Smale theory: Palais-Smale decomposition lemma, indexes of domains, Palais-Smale values, and Palais-Smale conditions. In section 2, we present some classical theorems such as the Lebesgue dominate convergence theorem, the Vitali convergence theorem and the compact imbedding theorems. We also present important lemmas such as Brezis-Lieb lemma and many lemmas for Sobolev spaces and Palais-Smale sequences which will be used for later sections. In section 3, we describe Palais-Smale decomposition theorem. In section 4, we prove that the four important Palais-Smale values are the same. Any one of them is called the index of J in Ω denoted by α(Ω) . We call that a nonzero solution u of equation (1) is a ground state solution if J(u)=α(Ω), and is a higher energy solution if J(u)>α(Ω) . We get a sufficient and necessary condition for that (PS)-condition holds. In section 5, We prove that if Ω is a large domain of R^N and if βis a positive (PS)-value for J, then mβ is also a (PS)-value in for J, where m=1,2,...(see Lemma 46).As a consequence, if Ω is an Esteban$-$Lions large domain, the set of all positive (PS)-values for J consists of mβ, where m=1,2,....(see Theorem 51).