Solutions Of Nonlinear Schr(?)dinger Systems

In this paper,we mainly study the existence of solutions for nonlinear k-coupled Schr(?)dinger equations.For nonlinear k-coupled Schr(?)dinger equations.we use vector form to express the elements,construct a product workspace of k times.and then we can define the equivalent inner product in space,then define equivalent norm,and get the corresponding energy functional.The existence and related properties of nontrivial solutions of k-coupled nonlinear Schr(?)dinger equations are studied by using the elliptic equation theory,such as the deformation lemma and the Nehai manifold.In the sub critical case,we analyzed the mountain pass theorem related to the general natrure of the abstract theorem proof and applied to the nonlinear k-coupled Schr(?)dinger equations.We obtain the result that when ?j>0,?ji ? 0.3ji = 3ij and matrix is a positive definiie matrix,the nonlinear k-coupled Schr(?)dinger equations have a nontrivial solution.In the critical case,we use the improved Nehai manifold method to prove the existence of the positive ground state solution of a general nonlinear k-coupled Schr(?)dinger system under certain conditions.We obtain the nonexistence results of nontrivial solutions by using prior estimates.