Klein-Gordon-Maxwell系统解的存在性和多解性

Existence and Multiplicity of Solutions to a Class of Klein-Gordon-Maxwell System

Klein-Gordon-Maxwell系统具有很强的物理背景,它提供了带电粒子物质和它所产生的电磁场之间作用的“二元模型”描述.根据这个模型,粒子物质是一个非线性场方程的孤波解,且电磁场的作用是由场方程与麦克斯韦方程耦合的衡量电位描述的.本文利用变分方法和临界点理论研究一类Klein-Gordon-Maxwell系统解的存在性和多重性.首先,利用山路引理,我们证明了系统非平凡解的存在性,其中一个解是非负的,一个解是非正的.其次,运用喷泉定理,文中证明系统在非线性项满足一定条件下无穷多高能量解的存在性.本文所得结果推广了以前的结论.

The Klein-Gordon-Maxwell system has strong physical backgrounds it can describe the“binary model”between the charged particle matter and the electromagnetic field it produces.According to this model,the particle matter is the solitary wave solution to a nonlinear field equation,and the effect of the electromagnetic field is determined by the coupling of the field equation with the Maxwell equation.In this paper,we use the variational method and critical point theory to study the existence and multiplicity of solutions for a class of Klein-Gordon-Maxwell systems.We first investigate the existence of non-trivial solutions to the above system by using mountain pass lemma,one of the solution is non-negative and the other one is non-positive.Secondly,under some assumptions on the nonlinear term,we establish the existence of infinitely many high energy solutions by using the fountain theorem.Our results generalize the previous conclusions.