The Existence And Multiplicity Of Solutions For The Schr(?)dinger-maxwell Equation

In recent decades,with the help of the modern variational methods,a large number of scholars have obtained the existence,non-existence,multiplicity and a series of results of solutions for Schr6dinger-Maxwell equation through making hypotheses on the potential function V and the nonlinear term f.Based on the existing literatures,the existence and multiplicity of solutions of three different types of Schrodinger-Maxwell equation are obtained by appropriately adding and weakening the potential function and the nonlinear term and using the critical point theory and variational method.According to the content,this paper is divided into the following five chapters:The first chapter is the introduction,which mainly introduces the background and significance of this topic,the research status,and some basic definitions and related lemmas.The second chapter focuses on the existence of infinitely many nontrivial solutions for the nonlinear Schrodinger-Maxwell equation by using the variational method and fountain theorem.The prior literature considered such as K(x)= 1 or K(x)= 0,f(u)= u,O<p<1 or 2<p<6,the novelty of this chapter is that it considers the situation that the existence of infinitely many nontrivial solutions for a kind of nonlinear Schrodinger-Maxwell equation can be obtained when K(x)and f(u)are a positive continuous function and 1<p<2.In the third chapter,we consider the existence of infinitely many high energy solutions for a kind of Schrodinger-Maxwell equation with sign changing potentials by using the critical point theory.The novelty of this chapter is that we consider that the potential function V is sign-changed and f(u)is a positive continuous function,and at the same time,the existence of infinitely many high energy solutions are obtained by weakening the partial restrictions of the function g(x,u).In the fourth chapter,we mainly study the existence of the ground state solutions for the following Schrodinger-Maxwell equations with periodic potential by using the Nehari manifold and mountain pass theorem.When 0<p<1,the problems in the solutions for Schrodinger-Maxwell equations,especially for its ground state solutions are more complex and it is more difficult to gain the existence of the solutions.The novelty of this chapter is considering the existence of ground state solutions when 0<p<1.The fifth chapter is a summary and outlook,and the author think the future research maybe can shift focus to weakening or improving some presumptions,for example,if there is no the limitation of AR conditions,can the existence and multiplicity of solutions for the Schrodinger-Maxwell equations still be obtained.