Abstract:

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.

Key words: Klein-Gordon-Maxwell system, mountain pass lemma, fountain theorem, high energy solutions