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

Abstract

<p style='text-indent:20px;'>In this paper, we consider the following Schrödinger-Poisson system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ p\in [4,6) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ a(x)\ge \lim_{|x|\to\infty}a(x) = a_{\infty}&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \lim_{|x|\to\infty}K(x) = 0 $\end{document}</tex-math></inline-formula>. Lack of any symmetry property of <inline-formula><tex-math id="M4">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ K $\end{document}</tex-math></inline-formula>, we present some new sufficient conditions to guarantee the existence of a positive ground state solution of above system. Our results extend and complement the ones of [G. Cerami, G. Vaira, J. Differential Equations 248 (2010)] in which <inline-formula><tex-math id="M6">\begin{document}$ p\in (4,6) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ K $\end{document}</tex-math></inline-formula> need to satisfy additional integrability conditions.</p>