Journal of the
Korean Mathematical Society JKMS

This paper is concerned with the following Klein-Gordon-Maxwell system: $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda V(x)u-(2\omega+\phi)\phi u=f(x,u), \mbox{ \ \ }x\in\mathbb{R}^{3},\\ \Delta\phi=(\omega+\phi)u^{2},\mbox{ \ \ }x\in\mathbb{R}^{3}, \end{array} \right. $$ where $\omega>0$ is a constant and $\lambda$ is the parameter. Under some suitable assumptions on $V(x)$ and $f(x, u)$, we establish the existence and multiplicity of nontrivial solutions of the above system via variational methods. Our conditions weaken the Ambrosetti Rabinowitz type condition.

Keywords: Klein-Gordon-Maxwell system, Sobolev embedding, variational methods, infinitely many solutions

MSC numbers: 35J20, 35B38