Existence Of Positive Solutions And Sign-Changing Solutions For Nonlocal Elliptic Equations

In this paper,we study the existence of positive solutions for fractional Schr(?)dinger-Poisson system and the existence of nodal solutions for Kirchhoff type equations by using the variational method.The main research content is as follows:In the first chapter,we summarize the background and present situation of the research and the main results.Chapter two is devoted to studying the existence of the positive solutions for fractional Schr(?)dinger-Poisson system(?)(1)where s,t ?(0,1),?>0,2<p<4,K(x),a(x)and b(x)are nonnegative functions satisfying some suitable conditions.The existence of positive solutions is proved by the refinement of Nehari manifolds and the compactness-concentration principle.Chapter three is devoted to studying the existence of the nodal solutions for Kirchhoff type equations-(a+b?R3|?u|2)?u+u=Q(|x|)|u|p-2u,x?R3,(2)where a,b>0,2<p<4,Q(|x|)is a nonnegative functions satisfying some suitable conditions.We definite nodal Nehari manifold,and establish the existence of nodal solutions by using energy estimates and Palais-Smale sequence.In chapter four,we summarize some prospects of the problems studied.