Radial Symmetry And Regulariy Of Solutions For Fractional Laplacian Equations

In this thesis, we discuss the radial symmetry for positive solutions of a Schr¨odinger typesystem with the fractional Laplacian. And we study the regularity of nonlinear equations forfractional Laplacian. This thesis is divided into three chapters.In chapter1, we introduce the background of the problem and main results of the thesis.Chapter2is concerned the radial symmetry of positive solutions for the following Schr¨odingertype system with the fractional Laplacianwhere0<α <1, p, q>1, N≥2. We are able to establish the radial symmetric theorem for thosepositive solutions by means of the moving plane method.In chapter3, we investigate the regularity of the nonlinear equation for fractional LaplacianwhereΩ R^N, N≥2, is a smooth bounded domain,0<s <1. We prove that any H^s（Ω） solutionu of problem belongs to L^∞（Ω） for the nonlinearity of f（t） being subcritical and critical. Thisimplies that the solution u is classical if f（t） is C^1,γ for some0<γ <1.