Variational And Topological Methods For Second Order Elliptic Boundary Value Problems

The aim of this thesis is to study the existence of weak solutions for semilinear second order elliptic boundary value problems under suitable conditions through topological and variational methods.In Chapter 1, we are devoted to introducing the development of topological degree theory and variational methods, also presenting the problems that will be studied.In Chapter 2, we introduce some essential definitions, preliminary theorems related to this thesis, and a way of reducing a kind of second order elliptic boundary value problems to finding the fixed points of a operator or the critical points of a corresponding functional.In Chapter 3, we introduce the settings of investgation of second order elliptc BVPS by variatioanl methods, we systematically introduce some classical sovability results obtained by the fixed point theorems, the least action principle and minmax methods in variational methods.In chapter 4, Under the generalised Ahmad-Lazer-Paul conditions, we obtain some sovability results of a kind of elliptic boundary value resonant problems by linking form critical point theorems, especial saddle point theorem, also present some other sovability conditions. We also obtain the nontrivial solutions for the problems by (generalised) mountain pass lemma. Furthermore, we give some conditions which can guarantee that the generalised Ahmad-Lazer-Paul conditions hold.