The Properties Of Solutions For Several Nonlinear Schr(?)dinger Systems With Fractional Laplacian

In this paper,we mainly study the properties of solutions for three different nonlin-ear Schrodinger equations with the fractional Laplacian by a direct method of moving planes.Using the integral defining the fractional Laplacian,we first obtain the key in-gredients needed in the method of moving planes for different types,such as maximum principle for anti-symmetric functions,narrow region principles and decay at infinity.For the semilinear elliptic systems We prove the positive solutions do not exist in the subcritical case:1<p,q?n+?/n-?besides p=q=n+?/n-?,and the solutions are radially symmetrical in the critical case:p=q=n+?/n-?.For the coupled nonlinear Schrodinger system with two equations and the nonlinear Schrodinger equations with three wave We prove the symmetry and mono tonicity of solutions.