Radial Symmetry And Liouville Type Theorem Of Nonnegative Solutions Of Some Nonlinear Fractional Equations And System

Fractional Laplacian is a nonlocal pseudo-differential operator which is defined in Rn.It can be used to model diverse physical phenomena,such as turbulence and water waves,anomalous diffusion,quasi-geostrophic flows and posudo-relativistic boson stars.It also has various applications in probability and finance.In this doctoral thesis,we consider some properties of nonnegative solutions and corresponding Liouville type theorem for some nonlinear fractional Laplacian equations and system.We also investigate some properties of ?-harmonic function with an isolated singularity.Firstly,we introduce the related knowledge of fractional Laplacian operator and the research background and development status.We also explain the content and main con-clusions of this study and give several lemmas to be used in the proof of theorems in the following chapters.Secondly,we consider the properties of nonnegative solutions for a fractional Lane-Emden type equation in an unbounded parabolic domain in Rn.Respect to the fractional Lane-Emden type equation,there were some results in the whole space Rn and the upper half space R+n,but there are no relevant conclusions in the parabolic domain.Instead of using the conventional extension method of Caffarelli and Silvestre or the method of moving planes in integral forms,a new method is employed to consider the monotonicity and symmetry of nonnegative solutions for the fractional Lane-Emden type equation.We prove that the nonnegative solutions are monotonically increasing about some component.As its application,we obtain the Liouville type results under proper conditions.Moreover,though making Kelvin transformation with the positive solution,we show that the positive solution is radially symmetric about the origin in Rn-1 both in the critical case and in the subcritical case.Thirdly,we investigate the properties of positive solutions for the fractional Henon equation with disturbance both in Rn and in R+n.We make Kelvin transformation with the positive solution and employ the method of direct moving planes to prove that the positive solution is radial symmetric about origin both in the subcritical case and in the critical case in Rn.Furthermore,we establish the corresponding Liouville-type theorem in the upper half space Rn+ under decay condition.We also obtain the symmetry of positive solutions both in the critical case and in the subcritical case in the upper half space R+n.Fourthly,we take into account the properties of positive solution of a nonlinear Schrodinger system with fractional diffusion in an unbounded parabolic domain,unite ball and the whole space Rn respectively.We first establish a narrow region principle for our system in the parabolic domain.Using this principle and the idea of the direct method of moving planes,we obtain the monotonicity about some component of positive solutions and the Liouville-type result for the nonlinear Schrodinger system with fractional diffusion.Moreover,according to the idea of the direct moving planes and the narrow region princi-ple,we obtain the radially symmetric result of positive solutions for the system in the unit ball when A(x)and B(x)are constants.In the end,we get the decay at infinity principle and the radially symmetric result of positive solutions for the system in the whole space when A(x)and B(x)are constants.Finally,we explore some properties of L-harmonic functions with an isolated singu-larity in Rn+1 and the admissible ?-harmonic functions with an isolated singularity in Rn.First of all,we introduce the definitions of L-harmonic function and the admissible a-harmonic function;afterwards,we deduce some Liouville results about the extension of the admissible a-harmonic function;in the next part,we establish the decomposition the-orems about the extension of the admissible ?-harmonic function in any ball and general region respectively;in the end,we extend the classical Bocher's Theorem to ?-harmonic functions with an isolated singularity in Rn.