The Existence Of Ground State Solution To A Fractional Nonlinear Schrodinger Equation

In this paper we study the existence of ground state solution to the following frac-tional nonlinear Schrodinger equation, with 0<s<1 and N>2s. The functions f(x,u) and V(x) satisfy some assumptions respectively. We consider the following two cases.In the first case:the equation (1) satisfies that V(x) are periodic and f(x, u) is non autonomous. With V:RNâ†'R and f:RN×Râ†' R are periodic. Fractional laplace operator can be portrayed as F((-â–³)sv)(ζ)=|ζ|2sF(v)(ζ).With F on be-half of the Fourier transform. We obtain the existence of ground state solution to the equation by using Nehari manifold method and critical point theory.In the second case:By applying the variational method we obtain that the ex-istence of weak solutions to the equation (1) when V(x) is bounded and f(x, u) is autonomous.This thesis consists of three chapters. The first chapter focuses on discussing the introduction including research background and prerequisite knowledge. The second chapter deals with the existence of the solutions for the equation under the first case and the main conclusion is Theorem 2.1.1. In the last chapter we study the existence of the solutions to the equation in the second case and the main result is Theorem 3.1.1.