Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential

In this paper, we consider the following Schrödinger-Poisson system\begin{document}$ \begin{equation*} \left\{ \begin{array}{ll}-\triangle u+u+K(x)\phi(x)u = a(x)|u|^{p-2}u, \ \ \ \ x\in { \mathbb{R}}^{3},\\-\triangle \phi = K(x)u^2, \ \ \ \ x\in { \mathbb{R}}^{3}, \end{array}\right. \end{equation*} $\end{document}where \begin{document}$ p\in [4,6) $\end{document} , \begin{document}$ a(x)\ge \lim_{|x|\to\infty}a(x) = a_{\infty}>0 $\end{document} and \begin{document}$ \lim_{|x|\to\infty}K(x) = 0 $\end{document} . Lack of any symmetry property of \begin{document}$ a $\end{document} and \begin{document}$ K $\end{document} , we present some new sufficient conditions to guarantee the existence of a positive ground state solution of above system. Our results extend and complement the ones of [G. Cerami, G. Vaira, J. Differential Equations 248 (2010)] in which \begin{document}$ p\in (4,6) $\end{document} , \begin{document}$ a $\end{document} and \begin{document}$ K $\end{document} need to satisfy additional integrability conditions.