Existence Of Solutions For Linearly Coupled Fractional Laplace Equations With Critical Exponent

In this paper,we mainly study the existence of solutions for the following system.(?)(-?)s is the fractional Laplace operator,0<s<1,?1,?2>0,2*=2N/N-2s,N>2s,??R.? is a smooth bounded open set on RN,?1,s(?)is the first eigenvalue of((-?)s,H0s(?)),?1,?2>-?1,s(?).In this paper,it is proved that the system has a positive ground state solution for 0<?<(?)by means of variational method,and it is also proved that the system has a positive high energy solution for ? which is small enough by using the perturbation method,topological degree and pseudo gradient vector field.At the same time,the asymptotic behavior of positive ground state solution and high energy solution for ??0 are also analyzed.The main contents of this paper are as follows:The first chapter introduces research background,research status,main conclusions and research ideas of the system.The second chapter introduces the basic knowledge for the solutions,including some definitions,energy functional,minimum energy value,several important lemmas and their proofs.In the third chapter,we prove that the system has a positive ground state solution by using the classical mountain pass theorem.At the same time,the asymptotic behavior of the positive ground state solution is investigated when ??0.In the forth chapter,by using topological degree and pseudo gradient vector field,it is proved that the system has a positive high energy solution.The asymptotic behavior of the positive high energy solution is also studied when ??0.