Some Properties For Solutions Of Fractional P-Laplace Equations

We investigate some properties for positive solutions of two fractional differential equations,including the fractional Schr?dinger equations and the fractional p-Laplace equations.We are concerned with the existence,symmetry and monotonicity of their solutions,and some symmetry results of both the solutions and domains.In Chapter 1,we introduce the history and significance of partial differential equations and the development and application of fractional differential theory in recent years.We also summarize some current research status about fractional Laplace equations and fractional p-Laplace equations,and give some mainly related properties about solutions of these equations.In Chapter 2,we introduce some well-known results for fractional Laplace equations and fractional p-Laplace equations,that is,the definitions,the Sobolev space and fractional Sobolev space,and some lemmas to be used in the subsequent proof such as fractional Hardy-Littlewood-Sobolev?H-L-S?inequality and Sobolev inequality.At last,we also give the definition and some properties of the Schwarz symmetric rearrangement.In Chapter 3,we investigate a class of fractional Schr?dinger equations in bounded annular domains.We transform the fractional Schr?dinger equations into a system of integral equations involving Bessel potentials and Riesz potentials.Then via the methods of moving planes and H-L-S inequality,we obtain some symmetry results for the solutions and domains.In Chapter 4,we investigate an overdetermined problem for fractional p-Laplace equations.The radial symmetry of both bounded domains and positive solutions are given by the method of moving planes under constants Dirichlet conditions and regularity boundary conditions.In Chapter 5,we investigate a class of fractional Schr?dinger p-Laplace equations.We obtain some existence and symmetry results for solutions in fractional Sobolev spaceW^s,p?R ⁿ?by rearrangement of its corresponding constrained minimization.