Properties Of Solutions Of Several Equations And Systems With Fractional Laplacian

This paper mainly studies symmetry of solutions of several equation and system-s with fractional Laplacian and the method of moving planes is the mainly research method.The paper is divided into four chapters.Chapter 1 introduces some background knowledge and conclusions in this paper.In Chapter 2 we consider the symmetry of the following fractional Laplacian equation with negative exponent in a unit ball:?-??^?/2u?x?=u^-p?x?.In Chapter 3 we study the symmetry of a coupled system with fractional Laplacian in the whole space:???In Chapter 4 we consider the symmetry of a fully nonlinear local system with fractional Laplacian in the whole space:???It is direct and effective for the method of moving planes to studying the symmetry of solutions in the unit ball of Rⁿand the whole space Rⁿ.The”maximum principle of anti-symmetry functions”,”narrow region principle”and”the decay at infinity”will play key roles in the method of moving planes.