Existence And Multiplicity Of Solutions For Two Nonlocal Elliptic Equations

In this paper,we mainly study the existence of nontrivial solutions for two types of nonlocal elliptic equations(Kirchhoff equation and Schr(?)dinger-Poisson system).For the Kirchhoff equation,we mainly consider the existence and multiple solutions in high dimensions(N?4)and we also consider the existence of bound state solutions with 2<p<4 for the Schr(?)dinger-Poisson system.This paper is divided into the following four chapters:In Chapter ?,we mainly introduce the research background and meaning of Kirchhoff equation and Schr(?)dinger-Poisson system;then give the research status and progress at home and abroad;finally,some definitions and basic theorems will be given and we shall introduce the structure arrangement of this paper.In Chapter ?,we consider the following non-autonomous Kirchhoff equations:where N?4,?>0,M(t)=at+b(a,b>0),2<p<2*=2N/(N-2).And here we require that Q?C(RN,R+)be non-negative continuous function and has k maximum points.By using a new constrained manifold method,we show that the above problem admits at least k positive solutions for N=4;while admits at least 2k positive solutions for N? 5.In Chapter ?,we consider the following non-autonomous Schr(?)dinger-Poisson systems:with 2<p<4.Under suitable assumptions on the decay rate of K(x)and a(x),but not requiring any symmetry property.We will prove that the existence of a bound state solution by developing a new constraint method together with the linking theorem.In chapter ?,we summarize the main results and raise questions which is worth exploring.