The Existence Of Solutions For The Schr(?)dinger-Maxwell Equations

Since the pioneering work[11]of V.Benci and D.Fortunato,based on variational methods and critical point theory,many scholars have studied the existence of results for Schrodinger-Maxwell equations.Including existence,multiplicity and nonexistence of nontrivial solutions;Existence of semi-classical solutions;Existence of ground state solutions;Ground state sign-changing solutions.In this paper,based on the existing literature,weaken the nonlocal or nonlinear term part of the constraints,we prove existence of solutions for Schrodinger-Maxwell equations.The proof is based on variational methods and critical point theory.Specific content as follows:In chapter 1,the development of variational methods and critical point theory are briefly introduced.The main work and basic knowledge of this paper also are involved.In chapter 2,the existence of infinitely many small solutions for nonlinear Schrodinger-Maxwell equations is studied by using the variable fountain theorem.Most literatures consider the situation when K(x)= 1,f(x,u)= 0,and g(x,u)was autonomous.The innovation of this chapter is that we only need K(x)to be a positive continuous function,f(x,u)to be a continuous function satisfying certain conditions,and g(x,u)to be non-autonomous.We investigate the existence of infinitely many small solutions in this case.The chapter 3,on the basis of chapter 2,further studies the following conclusion of the multi-solutions for nonlinear Schrodinger-Maxwell equations.ifferent from chapter 2,we study h(u)meet(?)u?R,(?)??0,2H(u)?h(u)u,and the onlinear term contains parameters.n chapter 4,we study the existence of ground state solutions for nonlinear Schrodinger-axwell equations,whose main tools are Nehari manifold and mountain road lemma,when g(x,u)=(p+1)b(x)|u|p-1u.The novelty of this chapter is that when 3<p<5,h(u)=u and g(x,u)to be nonautonomy is simultaneous.We concern with the existence of ground state solutions of the equations.Chapter 5 is the summary and reflection of this paper.It is further considered whether some assumptions can be weakened or improved,such as whether V(x)meets the periodicity or V(x)is allowed to be sign-changing,and the growth condition of g(x,u)and whether the solutions of these kinds of Schrodinger-Maxwell equations can still exist.