Radial Symmetry Of Positive Solutions For A Type Of Fractional Elliptical Equation And Lane-Emden System

In this paper, we mainly study radial symmetry and non-existence of positive solutions for a type of fractional elliptical equation and Lane-Emden system by the direct method of moving planes.In chapter one, we give a brief introduction of the research background for fractional Laplacian and the motivation of our problems.In chapter two, we mainly introduce the preliminary knowledge of fractional Laplacian.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. Finally,we present various maximum principles which will be commonly used in applying the direct method of moving planes.In chapter three, we study the properties of positive solutions for a type of fractional elliptical equation both in the whole space Rn and the upper half space R+n. Firstly, we prove the radial symmetry of positive solutions with the decay condition near infinity in Rn, and then prove radial symmetry and non-existence of positive solutions without the decay condition in Rn. Finally, we prove the non-existence of positive solutions in the half space R+n.In chapter four, we study the properties of positive solutions for the fractional Lane-Emden system. First, we establish the narrow region principle for system, and then apply the direct method of moving planes to the system to obtain the radial symmetry and non-existence of positive solutions.