Radial Symmetry And Monotonicity Of Positive Solutions For Fractional Laplace Equation

In this paper,we mainly study radial symmetry and monotonicity of positive solutions for a type of fractional Laplacian equation and fractional p-Laplacian equation systems by the direct method of moving planes,then we give a brief introduction the gradient estimate of f-Laplacian equation.In chapter one,we give a brief introduction of the research background for fractional Laplacian,fractional p-Laplacian and fractional f-Laplacian equation and the motivation of our problems.In chapter two,we mainly introduce the preliminary knowledge of fractional Lapla-cian and the proof of relevant conclusions.We first give some definitions of fractional Laplacian,and then introduce two old methods in studying it:the extension method and the equivalent integral equations method.Then,we present various maximum principles which will be commonly used in applying the direct method of moving planes.Finally,we applying these maximum principles to proof radial symmetry and monotonicity of positive solutions for fractional Laplacian with the singular nonlinearity.In chapter three,we mainly introduce the preliminary knowledge of fractional p-Laplacian and the proof of relevant conclusions.We first give some definition of fractional p-Laplacian.Then,we present various maximum principles which will be commonly used in applying the direct method of moving planes.Finally,we applying these maximum principles to proof radial symmetry and monotonicity of positive solutions to a system involving fractional p-Laplacian in a ball.In chapter four,we mainly introduce the basic knowledge of fractional f-Laplacian.Then study the gradient estimates for a nonlinear elliptic equation on a smooth metric measure space.