Existence Of Symmetry Solutions For Two Fractional Laplacian Equations With Critical Nonlinearity

In this paper we study the symmetry solutions of two fractional Laplacian equa-tions.The first problem is fractional Laplacian equations in the case of subcritial Where N> 2s, s E (0,1). By using Nehari manifold and variational methods proves the existence of solutions for the equation.We can find critical point in the sobolev space composed by radially symmetric funtion. The critical point is the solution for the equation.The second problem is fractional Laplacian equations in the case of critial Where N>2s, s ∈ (0,1) is fixed,2*(s)= 2N/(N-2s) is a fractional critical sobolev exponent.We can find critical point in the sobolev space composed by radi-ally symmetric funtion. And by using variational method including Mountain pass Theorem.We prove the existence of radially symmetric solutions for the equation.This thesis consist of three chapters. The first chapter is devoted to discuss the introduction including research background and prerequisite knowledge. The second chapter deals with the existence of the radially symmetric solutions for the equation under the first category, the main conclusion is Theorem 2.1.1. In the last chapter we research the existence of the radially symmetric solutions for the equation about the second category, the main results are Theorem 3.1.1.