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} -Δu+ V(x)u-(2ω+ϕ)ϕu=f(x,u)+h(x)&\mbox{in ${\mathbb{R}}^3$},\\ -Δϕ+ ϕu^2=-ωu^2&\mbox{in ${\mathbb{R}}^3$}, \end{array} \right. \end{eqnarray*} where $\omega >0$ is a constant, $u$, $\varphi :{\mathbb{R}}^3\to \mathbb{R}$, $V:{\mathbb{R}}^3\to \mathbb{R}$ 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 $h\equiv 0$ for above equations.