This paper is dedicated to studying the following Schrödinger–Poisson system -▵u+V(|x|)u+λϕu=f(|x|,u),x∈R3,-▵ϕ=u2,x∈R3,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u+V(|x|)u+\lambda \phi u=f(|x|,u), &{}\quad x\in \mathbb {R}^3,\\ -\triangle \phi = u^2,&{}\quad x\in \mathbb {R}^3, \end{array} \right. \end{aligned}$$\end{document}where λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} is a positive parameter, V∈C(R3,(0,∞))\documentclass[12pt]{minimal}
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\begin{document}$$V\in {\mathcal {C}}(\mathbb {R}^{3}, (0,\infty ))$$\end{document} and f∈C(R3×R,R)\documentclass[12pt]{minimal}
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\begin{document}$$f\in {\mathcal {C}}({\mathbb {R}}^{3}\times \mathbb {R}, \mathbb {R})$$\end{document}. Using weaker conditions lim|t|→∞∫0tf(x,s)ds|t|3=∞\documentclass[12pt]{minimal}
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\begin{document}$$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s){\mathrm {d}}s}{|t|^3}=\infty $$\end{document} uniformly in x∈R3\documentclass[12pt]{minimal}
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\begin{document}$$x\in \mathbb {R}^3$$\end{document}, and f(x,τ)τ3-f(x,tτ)(tτ)3sign(1-t)+θ0V(x)|1-t2|(tτ)2≥0,∀x∈R3,t>0,τ≠0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left[ \frac{f(x,\tau )}{\tau ^3}-\frac{f(x,t\tau )}{(t\tau )^3}\right] {\mathrm {sign}}(1-t) +\theta _0V(x)\frac{|1-t^2|}{(t\tau )^2}\ge 0, \quad \forall \;\; x\in \mathbb {R}^3,\ t>0, \;\; \tau \ne 0 \end{aligned}$$\end{document}with a constant θ0∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$\theta _0\in (0,1)$$\end{document}, instead of the usual super-cubic condition lim|t|→∞∫0tf(x,s)ds|t|4=∞\documentclass[12pt]{minimal}
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\begin{document}$$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s){\mathrm {d}}s}{|t|^4}=\infty $$\end{document} uniformly in x∈R3\documentclass[12pt]{minimal}
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\begin{document}$$x\in \mathbb {R}^3$$\end{document}, and the Nehari type monotonic condition on f(x,t)/|t|3\documentclass[12pt]{minimal}
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\begin{document}$$f(x,t)/|t|^3$$\end{document}, we establish the existence of one radial ground state sign-changing solution uλ\documentclass[12pt]{minimal}
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\begin{document}$$u_\lambda $$\end{document} with precisely two nodal domains. Under the same conditions, we also prove that the energy of any radial sign-changing solution is strictly larger than two times the least energy; and give a convergence property of uλ\documentclass[12pt]{minimal}
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\begin{document}$$u_\lambda $$\end{document} as λ↘0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \searrow 0$$\end{document}. Our result unifies both asymptotically cubic and super-cubic cases, which extends the existing ones in the literature.
