The Application Of Direct Method Of Moving Spheres To The Study Of The System With Fractional Laplacian

In this paper,we mainly study the semi-linear equations involving the fractional Laplacian (?) by a direct method of moving spheres.First,we prove the maximum principle and the narrow region principle for anti-symmetric functions;then we classify the non-negative solutions of(*);at last,by proving the equivalence between the differential equations (?) and the integral systems (?) we obtain the Liouville type theorem on the half space for the differential equations(**).