Multiple solutions for superlinear Klein-Gordon-Maxwell equations

Abstract

In this paper, we consider the following Klein-Gordon-Maxwell equations \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+ V(x)u-(2\omega+\phi)\phi u=f(x,u)+h(x)&\mbox{in $\mathbb{R}^{3}$},\\ -\Delta \phi+ \phi u^2=-\omega u^2&\mbox{in $\mathbb{R}^{3}$}, \end{array} \right. \end{eqnarray*} where $\omega>0$ is a constant, $u$, $\phi : \mathbb{R}^{3}\rightarrow \mathbb{R}$, $V : \mathbb{R}^{3} \rightarrow\mathbb{R}$ is a potential function. By assuming the coercive condition on $V$ and some new superlinear conditions on $f$, we obtain two nontrivial solutions when $h$ is nonzero and infinitely many solutions when $f$ is odd in $u$ and $h\equiv0$ for above equations.