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

Schr(?)dinger-Maxwell equations are obtained by physicists when solving the nonlinear Schr(?)dinger equation and the interaction of an unknown electric field.Variational methods and critical point theory are powerful tools in studying Schr(?)dinger-Maxwell equations.In this paper,based on the previous literatures,improving the potential term condition,weakening the known condition,and adding the sublinear term condition.We obtain the existence of ground state solutions for Schr(?)dinger-Maxwell equations by using the mountain pass theorem respectively.According to the content,the paper is split into five sections.The first part,we introduce the equations,some basic knowledge and related lemmas.The second chapter,we consider the existence of ground state solutions for Schr(?)dinger-Maxwell equations.(?)Where,V and K are functions of x,under the assumptions of V,K,f and the conditions satisfied are weaker than the previous literature,we obtain the existence of ground state solutions by using the mountain path theorem,which generalizes high-energy solution of the previous literature.The third chapter,we study the existence of ground state solutions for Schr(?)dinger-Maxwell equations with periodic functions.(?)Where,V and K are periodic functions of x,and the nonlinear term f is non-autonomous and periodic function of x,the condition f satisfied is weaker than the previous literature.The existence of ground state solution of the system is obtained by using the mountain path theorem without compactness,which generalizes the existence and multiplicity of solutions of the previous literature.The fourth chapter,we consider the existence of ground state solutions of Schr(?)dinger-Maxwell equation with sublinear term.(?)Where,g(x,u)is sublinear growth of u.under the assumptions of V,K,f,g,we can still prove the existence of the ground state solution of the system by using the mountain path theorem,which generalizes the conclusion of 0<p<1 and the high-energy solution of the system in the existing literature.The fifth chapter summarizes the paper and gives the corresponding prospect.Under some assumptions weakened or improved,whether can we still get the existence of ground state solution of Schr(?)dinger-Maxwell equations.