Symmetry And Nonexistence Of Solutions For Nonlinear Equation And The System Of Equations Involving The Fractional Laplacian

In this paper we study radial symmetry and monotonicity of positive solutions for in the whole space Rn, where 0< ?< 2,?> 1. Furthermore, in R+n with the subcritical case and critical case 1<?_i+?_i??n+?2??/?n-??, we investigate the nonexistence positive solutions ofIn chapter one, firstly, we introduce the research background of the radial symmetry, nonexistence of the positive solution for fractional Laplacian and present some basic prop-erties and the motivation. Next, we simply introduce some important theorems and results which we shall use later.In chapter two, firstly, we recall some important definitions and theorems. Second-ly, we use the Maximum Principle and Liouville theorem to prove the fractional H???non equation ?0-3? is equivalent to the following integral equation u?x?= ?_Rⁿ G?x,y?|y|^?u^p?y?dy, x?Rⁿ, ?0-5? where ???is the Green's function of ?-??^?/2 in Rn. Lastly, by combining the method of moving planes in integral forms with a certain type of Kelvin transform, in the subcritical case, we obtain that positive solutions for fractional H???non equation ?0-3? are radially symmetric and monotone decreasing about the origin under some integrability condition.In chapter three, first, we use Kelvin transform and the method of moving planes in integral forms to prove ?0-4? is equivalent to the following integral equations with 1< ?_i??_i??n+?+2??/?n-??, where G?x,y? is the Green's function associated with ?-??^?/2 in R₊ⁿ. Then we continue work on integral systems ?0-6? to establish Liouville type theorems, i.e. the nonexistence of positive solutions in the subcritical case and critical case 1??i??i??n+?+2??/?n-??.