This paper is devoted to studying positive bound solutions for the following fractional Schrödinger–Poisson system \(\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+V(x)u=\phi |u|^{2_{s}^{*}-3}u+|u|^{2_{s}^{*}-2}u, & \quad x\in \Omega ,\\ (-\Delta )^{s}\phi =|u|^{2_{s}^{*}-1}, & \quad x\in \Omega ,\\ u=\phi =0, & \quad x\in {\mathbb {R}}^{3}\backslash \Omega , \end{array} \right. \end{aligned}\) where \(\Omega \subset {\mathbb {R}}^{3}\) is an unbounded exterior domain, \(\partial \Omega \ne \emptyset \) , \({\mathbb {R}}^{3}\backslash \Omega \) is bounded, \(s\in (\frac{1}{2},1)\) , \(2_{s}^{*}=\frac{6}{N-2s}\) is the fractional critical Sobolev exponent, and \(V\in L^{\frac{3}{2s}}(\Omega )\) is a non-negative function. Under the condition that \(|V|_{\frac{3}{2s}}\) and \({\mathbb {R}}^{3}\backslash \Omega \) are small enough in a prescribed sense, combining variational methods and the Brouwer degree theory, we derive that this equation has at least one positive bound state solution. Our result still holds true in the case \(\Omega ={\mathbb {R}}^{3}\) , hence, this paper extends and supplements some recent works about the Benci–Cerami problem for the fractional Schrödinger–Poisson system.
