Ground State Solutions for the Critical Klein-Gordon-Maxwell System

In this article, we study the following Klein-Gordon-Maxwell system involving critical exponent $$(\text{KGM})\begin{cases}-\Delta{u}+V(x)u-(2\omega+\phi)\phi{u}=\lambda\mid{u}\mid^{q-2}u+\mid{u}\mid^4u,\;\;\;\;x\in\mathbb{R}^3,\\-\Delta\phi+\phi{u}^2=-\omega{u}^2,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in\mathbb{R}^3,\end{cases}$$ where λ and ω are two positive constants. We found the existence of positive ground state solutions (that is, for solutions which minimizes the action functional among all the solutions) of (KGM) which improves some previous existence result in Carriao et al.(2012) [8].