Existence Of Solutions For A Class Of Schrodinger-Poisson Equations

In this paper, we study the existence of nontrivial solutions of a class of Schrodinger-Poisson equation via variational methods. Since this type equation arises from quantum theory and the theory of semiconductor, there is not only an extensive application background but also many challenging problems on the equation in the view point of mathematic.The paper consists of the following parts:In the first part, we summarize some background and the latest progresses on the study of Schrodinger-Poisson equation.The second part is preliminary tools in partial differential equation.In the third part, we study the autonomous Schrodinger-Poisson equation where μ>0. Under the assumption on f(u) just near the origin, we obtain the existence of nontrivial radial solutions and the dependence of the solutions on the parameter μ via variational method. We also extend the results to the case of a radial symmetry potential.In the fourth part, we study the following Schrodinger-Poisson equation. where μ>0. We assume that f(u) satisfies the local condition just near origin and V(x) is a well potential function, i.e.V(∞):=lim inf V(x)≥V(x) In addition to the basic ideas used in the third part, we apply concentration compactness lemma to overcome the lack of compactness and Moser iteration methods. The existences of nontrivial solutions of are obtained.