We study the Schrödinger–Poisson system: -Δu+u+λϕu=axup-2uinR3,-Δϕ=u2inR3,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi u=a\left( x\right) \left| u\right| ^{p-2}u &{} \text { in }{{\mathbb {R}}}^{3}, \\ -\Delta \phi =u^{2} &{} \ \text {in }{{\mathbb {R}}}^{3}, \end{array} \right. \end{aligned}$$\end{document}where parameter λ>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda >0$$\end{document}, 2<p<3\documentclass[12pt]{minimal}
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\begin{document}$$2<p<3$$\end{document} and ax\documentclass[12pt]{minimal}
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\begin{document}$$a\left( x\right) $$\end{document} is a positive continuous function in R3\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}^{3}$$\end{document}. Assuming that ax≥limx→∞ax=a∞>0\documentclass[12pt]{minimal}
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\begin{document}$$a\left( x\right) \ge \lim _{\left| x\right| \rightarrow \infty }a\left( x\right) =a_{\infty }>0$$\end{document} and other suitable conditions, we explore the energy functional corresponding to the system which is bounded below on H1R3\documentclass[12pt]{minimal}
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\begin{document}$$ H^{1}\left( {{\mathbb {R}}}^{3}\right) $$\end{document} and the existence and multiplicity of positive (ground state) solutions for Appa∞2/p-2<λ≤Appa12/p-2,\documentclass[12pt]{minimal}
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\begin{document}$$\left[ \frac{A\left( p\right) }{p} a_{\infty }\right] ^{2/\left( p-2\right) }<\lambda \le \left[ \frac{A\left( p\right) }{p}a_{1}\right] ^{2/\left( p-2\right) },$$\end{document} where Ap:=26-p/23-p3-pp-2p-2\documentclass[12pt]{minimal}
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\begin{document}$$A\left( p\right) :=2^{\left( 6-p\right) /2}\left( 3-p\right) ^{3-p}\left( p-2\right) ^{\left( p-2\right) }$$\end{document} and a∞<a1<amax:=supx∈R3ax.\documentclass[12pt]{minimal}
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\begin{document}$$a_{\infty }<a_{1}<a_{\max }:=\sup _{x\in {{\mathbb {R}}} ^{3}}a\left( x\right) .$$\end{document} More importantly, when ax=ax\documentclass[12pt]{minimal}
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\begin{document}$$a\left( x\right) =a\left( \left| x\right| \right) $$\end{document} and a0=amax,\documentclass[12pt]{minimal}
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\begin{document}$$a\left( 0\right) =a_{\max },$$\end{document} we establish the existence of non-radial ground state solutions.
