Schr(?)dinger Equations With Van Der Waals Type Potentials

In this paper,we consider the following elliptic problem:-?u+V(x)u+?(??*|u|p)|u|p-2u=(??*|u|q)|u|q-2u,x?R3,(0.0.2) where ?>0,0<?<3,0<?<3,1+?/3<p<3+?,q>1 and ??(x)=|x|?-3,??(x)=|x|?-3.If V=1,we show the existence of a ground state solution and multiple solutions,as well as the nonexistence result for(0.0.2).Furthermore,we investigate the limit behavior of ground state solutions of(0.0.2)as ??0+.Ground state solutions are also obtained if the potential V is not a constant.This thesis is divided into four chapters.In chapter 1,we introduce the background of the problem and main results of the thesis.In chapter 2,we give some preliminaries and obtain a Pohozaev identity to problem(0.0.2).In chapter 3,we consider the case that the potential ? is a constant,that is,V?1.First of all,we demonstrate the existence of a ground state solution of(0.0.2)by the constraint variational method.Furthermore,we explore the limiting behavior of ground state solutions as ??0+.Finally,multiple solutions are obtained.In chapter 4,using the concentration-compactness principle,we show the existence of ground state solution of(0.0.2)if the potential ? is not a constant.