The Solutions Of Coupled Schr(?)dinger System With Fractional Laplacian

In this paper, we mainly discussed the existence of the solutions of coupled Schr?dinger equations with nonlocal operator in a bounded domain and the entire space. Firstly, we give some conclusions about the solution w(x) of some Schr?dinger equation with fractional Laplacion operator respectively in a bounded domain and whole space. And then on the basis of these existing conclusions,we applied the variational method to research the existence of solutions of coupled Schr odinger equations with fractional Laplacian op-erator in a bounded domain and the whole space. The equations we studied are as follows: where ?(?)Rn,the nonlocal operator Aa is defined as follow:In the bounded domain, we mainly proved the existence of the nontrivial and least energy solution u(x)= (u1,u2) of equations (0.2) while 0??<min{?1,?2} or?> max{?1,?2} holds, and we also present the relationship between the solution u(x) with the solution w(x) we mentioned above.In the cases of entire space Rn(n? 3), we proved that there exist a constant /u0, such that the equation system exist a positive radial and least energy solution while 0??< /uo and N, ? satisfies Besides, we also showed that while min{?1,?2}??< ?(?)?2?1 holds, the least energy of the system can't be obtained; and while min{?1,?2}??? max{?1,?2}, the system doesn't have a nontrivial solution.