Existence Of Solutions To The Nonlinear Schr(?)dinger-Poisson Equations

This thesis for Master's degree considers existence of solutions of the two nonlinear Schrodinger-Poisson equations.The whole paper is arranged as follow:The chapter 1 is an introduction. We mainly consider the background and the recent development of the problems.In chapter 2, We give preliminary.We collect some basic materials in partial differential equation and some important theorems which will be used in the next chapters.In chapter 3, we consider the first equation, when V and K are positive con-stants, by restricting the functional to the radial functions where compact embed-dings hold. Then the problem has a ground state solution (u,Φ) for any(?)< q≤2 and p∈(2,5). If V is a non-constant potentials and K is a constant,we assume that V satisfy some certain conditions. We will use an alternative method. We prove that the establishment of compactness.Then the problem has also a ground state solution (u,Φ) for any(?)< q≤2 and p∈(3,5). In chapter4, we consider the second equation,λ> 0 is a parameter, V and K maybe non-radial functions.We assume that V and K satisfy some certain con-ditions.We use the Mountain Pass Theorem to find a solution to the equation. We prove that the problem has a positive solution forλsmall and has only a trivial solution forλlarge.