Non-autonomous Schrödinger-Poisson system in <inline-formula><tex-math id="M1">\begin{document} $\mathbb{R}^{3}$\end{document}</tex-math></inline-formula>

• DOI: 10.3934/dcds.2018077
• Publication Date: 2018-01-16T02:46:20Z
• Authors: Juntao Sun; ; Tsung-Fang Wu; Zhaosheng Feng
• Keywords: 13. Climate action; 0101 mathematics; 01 natural sciences

We study the existence of positive solutions for the non-autonomous Schrodinger-Poisson system: \begin{document}$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$ \end{document} where \begin{document} $\lambda >0$\end{document} , \begin{document} $2 and both \begin{document} $K\left( x\right) $\end{document} and \begin{document} $a\left( x\right) $\end{document} are nonnegative functions in \begin{document} $\mathbb{R}^{3}$\end{document} , which satisfy the given conditions, but not require any symmetry property. Assuming that \begin{document} $% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$\end{document} and \begin{document} $\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$\end{document} , we explore the existence of positive solutions, depending on the parameters \begin{document} $\lambda$\end{document} and \begin{document} $p$\end{document} . More importantly, we establish the existence of ground state solutions in the case of \begin{document} $3.18 \approx \frac{{1 + \sqrt {73} }}{3} .