Existence Of Solutions For A Class Of Klein-gordon-maxwell Equations With Parameters

In this paper,we use variational method to study the existence of solutions for a class of nonlinear Klein-Gordon-Maxwell equations on whole space,and obtain the dependence of soultions on parameters.The main novelty is that the nonlinear term only needs conditions near the zero point in the equation,and it is no longer limited at infinity.By adapting the method of modification the nonlinear term,the mountain pass theorem is applied to the research of the problem,and obtain the existence results,which is an important improvement to the related results in literatures on the problem in the whole space.This paper is mainly divided into three parts.In the first part,we use the variational method and Ni's inequality to study the existence of solutions for the Klein-Gordon-Maxwell equation with the radial symmetry potential,and obtain the dependence of solutions on parameters.where ?>0,??R.In this part,we limit the whole space to the subspace Hr1(R3)={u?H1(R3)|u=u(|x|)} to overcome the lack of compactness in the whole space.Finally,we use a Ni's inequality to get the L? estimation for the solution.In the second part,we use mountain pass theorem and Moser iterative technique to study the existence of solutions for the Klein-Gordon-Maxwell equation under the coercive potential,and obtain the dependence of solutions on parameters.In this part,we limit the whole space to the weighted Sobolev space Hv1(R3)={u?H1(R3)|fR3V(X)u2<+?} to overcome the lack of compactness in the whole space.Finally,we use the Moser iterative technique to get the existence result of the solution.In the third part,we use mountain pass theorem to study the existence of solutions for the Klein-Gordon-Maxwell equation with periodic potential,and obtain the dependence of solutions on parameters.We use the translation invariance of the functional to obtain partial compactness.Finally,we use the Moser iteration technique to get the existence result of the solution.