The Research On Existences Problem Of Schrodinger-Poisson's Solution

In this thesis,using some variational method,we study the existence of solutions to two classes of elliptic equations.This thesis contains an introduction part and four chapters.In the introduction,we review the background and motivation of the problems,and introduce a variational framework.In the first chapter,some preliminaries are introduced.In the second chapter,the following Schr(?)dinger-Poisson equation with the pa-rameter ? is considered:where ? is a real parameter,K(x),a(x)are non-negative functions defined on R3.Assuming that and satisfying some other assumptions.Nehari's technique is used to prove the ex-istence of positive ground state solutions to the problem.The new element here is that we first prove the existence to ground state solution of the classical equation by using the basic three-step method(see lemma 2.2.7),then we use the new conditions(see theorem 2.3.1 and theorem 2.3.2),and simple calculation to obtain the existence solutions to the problem.In the third chapter,the following Schr(?)dinger-Poisson equation is considered:with p?(3,5).Assuming that a:R3?R are nonnegative functions satisfying and other assumptions.Using Nehari's technique and the linking theorem,the exis-tence of the positive bound state solution to the above equation under some weaker conditions.The innovation of this chapter is the proof of A>0(see lemma 3.3.2)and the application of basic analytical methods(see lemma 3.2.6)to the compactness problem.In the fourth chapter,we summarize the results of this thesis and discusses some possible further research work.