Existence of positive solution for Klein–Gordon–Maxwell system without subcritical growth and Ambrosetti–Rabinowitz conditions

This article concerns the following Klein–Gordon–Maxwell system $\left(\begin{array}{cc}-\Delta u+V\left(x\right)u-(2\omega +\varphi )\varphi u={\left|u\right|}^{s-2}u+\lambda f\left(u\right),& x\in R^3,\\ \Delta \varphi =(\omega +\varphi )u^2,& x\in R^3,\end{array}\right)$where $\omega >0$ is a constant, $4\le s<6$, $\lambda >0$ is a parameter. When $f$ only satisfies suplinear conditions but not satisfies subcritical growth and Ambrosetti–Rabinowitz conditions, the existence of positive solution can be proved via variational methods, Moser iteration and perturbation arguments. Our result unifies both critical or supercritical cases and generalizes and improves the existing ones.