Symmetry And Monotonicity Of Positive Solutions Of Some Fractional Laplace Equations

In this paper, we mainly study radial symmetry and monotonicity of positive solutions of some fractional Laplace equations in the unit ball by the method of moving planes. There are two principal methods in this paper:one is the method of moving planes in integral forms; the other is a direct method of moving planes.In chapter one, we first introduce the backgrounds of the fractional Laplacian studied in this dissertation. Next, we review some related research results and then elaborate the basic idea in this paper.In chapter two, we recall Hopf lemma and the specific form of Green's function in the unit ball and its properties, and then give the maximum principle of the fractional Laplacian. We establish the equivalence of solutions to the integral equation and the differential equation in the sense of distribution. Based on the equivalence, we will prove the radial symmetry and monotonicity of positive solutions to the integral equation by using the moving planes.In chapter three, we give the narrow region principle. Then we will apply a direct method of moving planes to singular positive solutions of some semi-linear equations and the Brezis-Nirenberg equation to obtain their symmetry and monotonicity.