On the existence and multiplicity of solutions for nonlinear Klein–Gordon–Maxwell system

In this paper, we study the existence and multiplicity solutions for the following Klein–Gordon–Maxwell system \begin{align*} \begin{cases} - \Delta u +V(x)u-(2\omega+\phi)\phi u =f(x,u), &x\in \mathbb{R}^3,\\ \Delta \phi =(\omega+\phi)u^2, \quad & x\in \mathbb{R}^3, \\ \end{cases} \end{align*} where $\omega>0$ is a constant and the nonlinearity $f(x,u)$ is either asymptotically linear in $u$ at infinity or the primitive of $f(x,u)$ is of 4-superlinear growth in $u$ at infinity. Under some suitable assumptions, the existence and multiplicity of solutions are proved by using the Mountain Pass theorem and the fountain theorem, respectively