Existence Of Solutions For Coupled System Of Elliptic Equation With Critical Exponent And Fractional Laplace Equation With Subcritical Exponent

In this paper,we consider the existence of solutions for the following coupled system of fractional Laplace equation with subcritical exponent and elliptic equation with critical exponent:(?)where(-?)s is the fractional Laplacian,s?(0,1),??RN,N?3;?>-?1,s(?),?>-?1(?),?1,s(?)is the first eigenvalue of((-?)s,H0s(?)),?1(?)is the first eigenvalue of(-?,H01(?));??R is a coupling parameter;2*=2N/N-2 is the Sobolev critical exponent;1<p<2s*-1,2s*=2N/N-2s.By variational method,we show that this system has a positive ground state solution for ?(>0)in some proper range.By a perturbation method,we prove that the system has a positive high energy solution for |?| small enough.Moreover,the asymptotic behaviors of the positive ground state and higher energy solutions as ??0 are analyzed.This paper is organized as follows:In the first chapter,we mainly introduce the research background and research status,then we introduce the main conclusions and research methods of this paper.In the second chapter,we introduce the important definition and key theorems,including the knowledge about fractional Sobolev space and variational methods.In the third chapter,we use the conclusions about two single equations and the Mountain Pass Theorem to prove that the system has a nontrivial positive ground state solution,and we study the asymptotic behavior of the ground state solution as ??0.In the fourth chapter,we construct a special P.S.sequence to prove the existence of positive high energy solution by the perturbation method.And we study the asymptotic behavior of the high energy solution as ??0.In the fifth chapter,we summarize the main research results of this paper and put forward some prospects for the research problems.