In this paper, we study a class of nonlinear Schrödinger–Poisson systems with indefinite steep potential well: -Δu+Vλ(x)u+K(x)ϕu=|u|p-2uinR3,-Δϕ=Kxu2inR3,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$\end{document} where 3<p<4\documentclass[12pt]{minimal}
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\begin{document}$$3<p< 4$$\end{document}, Vλ(x)=λa(x)+b(x)\documentclass[12pt]{minimal}
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\begin{document}$$V_{\lambda }(x)=\lambda a(x)+b(x)$$\end{document} with λ>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda >0$$\end{document} and K(x)≥0\documentclass[12pt]{minimal}
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\begin{document}$$ K(x)\ge 0$$\end{document} for all x∈R3\documentclass[12pt]{minimal}
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\begin{document}$$x\in \mathbb {R}^{3}$$\end{document}. We require that a∈C(R3)\documentclass[12pt]{minimal}
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\begin{document}$$a\in C( \mathbb {R}^{3}) $$\end{document} is nonnegative and has a potential well Ωa\documentclass[12pt]{minimal}
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\begin{document}$$\Omega _{a}$$\end{document}, namely a(x)≡0\documentclass[12pt]{minimal}
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\begin{document}$$a(x)\equiv 0$$\end{document} for x∈Ωa\documentclass[12pt]{minimal}
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\begin{document}$$x\in \Omega _{a}$$\end{document} and a(x)>0\documentclass[12pt]{minimal}
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\begin{document}$$a(x)>0$$\end{document} for x∈R3\Ωa¯\documentclass[12pt]{minimal}
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\begin{document}$$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$\end{document}. Unlike most other papers on this problem, we allow that b∈C(R3)\documentclass[12pt]{minimal}
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\begin{document}$$b\in C(\mathbb {R}^{3}) $$\end{document} is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as λ→∞\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \rightarrow \infty $$\end{document}.
