On A Class Of Fractional Schr(?)dinger System

In this paper,we discuss existence of nontrivial solution to the following class of fractional nonlinear Schr(?)dinger system:(??)s1u+V1(x)u = f1(u)+?(x)v,x ??,(-?)s2v+V2(x)v=f2(v)+?(x)u,x ??,u=0,v=0,x?RN\?.where,(-?)s denotes the fractional Laplacian,0<s1,s2?1,N?2.potentials V1 and V2 are positive functions which are bounded away from 0 in ?.coupling function A satisfies:|?(x)|??(?)for some ??(0,1).nonlinear terms fl and f2 are functions with subcritical growth.In this paper,we focus on the research about existence of nontrivial solution for the fractional Schr(?)dinger system when ? is bounded domain or RN.This paper is divided into five chapters as follows:In the first chapter,firstly it mainly introduces the background and research sta-tus about the fractional Schr(?)dinger system,then it introduces the main contents and conclusions of this paper.In the second chapter,it introduces the basic knowledge which is necessary for our problem,including the fractional Sobolev space,fractional Laplacian and variational methods.In the third chapter,firstly we assume that ? is a bounded domain,by using a variant of the Mountain Pass Lemma which considers the Cerami sequence,we get a Cerami sequence of our problem and prove boundness result of this sequence.Then by the compactness of Sobolev embedding on bounded domain,we get a nontrivial weak solution of our problem.In the fourth chapter,we assume that Q = RN.By the Concentration Compactness Principle,we establish a existence result of nontrivial solution for our problem.In the fifth chapter,we discuss some prospects of this paper,including the growth condition of nonlinear terms,the types of potential functions,the definition domain of equations and regularity of solutions.