Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

Abstract

<p style='text-indent:20px;'>In the present paper we study a class of Schrödinger-Poisson equations</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE0"> \begin{document}$\begin{equation} \begin{cases} -\Delta u+V(x)u+\phi u = a (x)|u|^{p-1}u,\ x\in\mathbb{R}^3\\ -\Delta \phi = u^{2},\ x\in\mathbb{R}^3, \end{cases} \end{equation}\quad\quad\quad (1)$ \end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ a(x) $\end{document}</tex-math></inline-formula> are of different forms on the half space, i.e. <inline-formula><tex-math id="M3">\begin{document}$ V(x) = V_{1}(x), a(x) = a_{1}(x) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}$ x_{1}>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ V(x) = V_{2}(x), a(x) = a_{2}(x) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ x_{1}<0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ V_{1},V_{2},a_{1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ a_{2} $\end{document}</tex-math></inline-formula> are periodic in each coordinate direction. By using a concentration compactness discussion, we establish the existence of surface gap soliton ground state of (1) for <inline-formula><tex-math id="M9">\begin{document}$ p\in [3,5) $\end{document}</tex-math></inline-formula>. We also give a Mountain-Pass type ground state of (1) for <inline-formula><tex-math id="M10">\begin{document}$ p\in (3,5) $\end{document}</tex-math></inline-formula>.</p>