Dichotomous Concentrating Solutions For A Schr(?)dinger–Newton Equation

In this paper,we mainly consider the following Schrodinger-Newton equation where ε is a positive parameter and V(x)is the potential function.Many previous studies have proved that the above Schrodinger-Newton equation has a solution that concentrates on a certain point,we prove that the solution of the above equation is non-degenerate.In addition,We demonstrate an interesting phenomenon,which we call dichotomy,for concentrating solutions of the above Schrodinger-Newton equation.More specifically,we show the existence of infinitely many concentrating solutions which concentrate both in a bounded domain and near infinity.The paper is divided into the following four chapters.In Chapter 1,we introduce the background of Schrodinger-Newton equation and the main results of this thesis.In Chapter 2,we introduce some preliminaries and necessary lemmas,give the proof of the Pohozaev identity and the decay of the solution,and then estimate the energy of the approximate solution.In Chapter 3,we prove that the solution of the above equation is non-degenerate by contradiction.In Chapter 4,we prove the existence of the dichotomous concentrating solutions by the method of Lyapunov-Schmit reduction.