In this paper we prove the existence of normalized solutions \((\lambda ,u)\subset (0,\infty )\times H^1(\mathbb {R}^3)\) to the following Schrödinger–Poisson equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+\lambda u+(|x|^{-1}*u^2)u=|u|^{p-2}u\quad \text {in}\quad \mathbb {R}^{3},\\ \displaystyle u>0,\quad \int _{\mathbb {R}^{3}}u^2dx=a^2, \end{array}\right. } \end{aligned}\) where \(a>0\) is fixed, \(p\in (\frac{10}{3},6)\) is a given exponent and the potential V satisfies some suitable conditions. Since the \(L^2({{\mathbb {R}}}^3)\) -norm of u is fixed, \(\lambda \) appears as a Lagrange multiplier. For \(V(x)\ge 0\) , our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al (Commun Partial Differ Equ 46:1729–1756, 2021). For \(V(x)\le 0\) , we directly construct an entire mountain-pass solution with positive energy.
