In this paper, we consider the following Schrödinger–Poisson system \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+u+\Phi (y)u=Q_n(y)|u|^{p-2}u, \hspace{3mm}y \in {\mathbb {R}}^3,\\ -\Delta \Phi (y) =u^2, \hspace{3mm}y \in {\mathbb {R}}^3, \end{array}\right. } \end{aligned}\) where \(p\in (4,6)\) and \(Q_n\) are concrete bounded functions whose self-focusing core \(supp\{Q_n^+\}\) shrinks to a finite set of points as \(n\rightarrow \infty \) . We prove the existence of positive ground state solution \(u_n\) corresponding to \(Q_n\) and study the limiting profile of \(u_n\) as \(n\rightarrow \infty \) . We also prove the existence of localized bound state solutions for above Schrödinger–Poisson system by using penalization method. Moreover, by using Green’s representation formula, we improve the result in [14, Theorem 1.3] for nonlinear Schrödinger equation.
