Existence Of Solutions For Klein-Gordon-Maxwell Equations

Through making appropriate assumptions on potential and nonlinear terms,this paper obtains the existence of solutions for the different types of Klein-Gordon-Maxwell equations by using Variational Methods and Critical Point Theory.According to contents,the thesis is divided into five chapters.Chapter 1 mainly introduces the paper's background,research methods,research status as well as related definitions,related lemmas and the whole structure of this paper.Chapter 2 studies the multiplicity of solutions for the following nonhomogeneous Klein-Gordon-Maxwell system by using Variational Methods.The most of previous literatures analyze the existence of solutions for the equations under the condition of a(x)? 1 and g(x)? 0,while this chapter studies the existence of multiplicity of such solutions under the situation that a(x),g(x)are positive continuous functions.Chapter 3 probes into the existence of infinitely many solutions for the following Klein-Gordon-Maxwell system with parameters by adopting Fountain Theorem and Dual Fountain Theorem.Introducing parameters will make the research on the existence of the infinitely many so-lutions for the Klein-Gordon-Maxwell equation become more complicated.And this paper adds nonlinearities after introducing two parameters and then obtains the existence of the infinitely many solutions for the Klein-Gordon-Maxwell equation.The novelty of this paper is that it does not require that a(x)? b(x)? 1 a(x)? 1 and b(x)? 0,and just needs a(x),b(x)are positive continuous functions.Chapter 4 applies Mountain Pass Theorem and Nebari manifolds to study the existence of ground state solutions for the following nonlinear Klein-Gordon-Maxwell system.This chapter analyzes the existence of ground state solutions for the system with a more simple way like using Nehari manifolds instead of Pohozaev identity.In addition,this chapter does not require a(x)? 1,b(x)= 0 or a(x)? b(x)? 1,and just needs that a(x),b(x)are continuous and function g is a binary continuous function about x,u.Chapter 5 summarizes the main work,and makes reflections on the shortcomings of this paper,and points out some places that should be improved on.