Ground-state Solutions For Schr(?)dinger-maxwell Equations

Schr?dinger-Maxwell equation is one of the most important equations in quantum m?-chanics,it is used to describe the interaction between charged particles in electromagnetic field.Variational method and critical point theory are the hot tools to study this equation.In this paper,based on previous studies,we study the existence of ground state solutions for three different types of Schr?dinger-Maxwell equations under some suitable assumption by using two variational type of Mountain Pass Theorem.According to the content of the study,the paper is divided into the following five chapters:Chapter 1 gives an overview of some relevant basic knowledge and background.In chapter 2,we study the existence of the ground state solution for the following sublinear Schr?dinger-Maxwell equations,while O ? p ? 1.In this system,K is a continuous function of x and f ?u the nonlinear term is not autonomous.In this chapter,we use variational method and critical point theory to prove the existence of ground state solutions of the above equations on the corresponding Nehari manifold.In chapter 3,we concern the existence of ground state solutions for the following Schr?dinger-Maxwell equations in the critical conditions,which the requirements of k and f are similar to those in chapter 2,and the nonlinear terms is critical grown.Based on previous studies,this chapter is using the mountain pass theorem in variational form and the approximation method to extends the work on second chapter to the critical case,and prove the existence of ground state solutions.Chapter 4 mainly discusses the existence of ground state solutions for the following Schr?dinger-Maxwell equation with two nonlinear terms,while? ? 0,? ? 0,p ?(1,3).On the basis of previous studies,the coefficient of(?)?K is a constant non-positive continuous function about x,and the nonlinear term is non-autonomous and critical growth.In order to prove that the corresponding functional satisfies(PS)c condition,a function of u? about x,and a function of yt about u? are constructed in this chapter.Finally,the existence of the ground state solution is found.The last chapter will draw a conclusion and make a prospect for the system of this paper.