We study the Schrödinger-Poisson-Slater equation $ \begin{equation*} \left\{\begin{array}{lll} -\Delta u + \lambda u + \big(|x|^{-1} \ast |u|^{2}\big)u = V(x) u^{ p_ \varepsilon-1 } , \, \text{ in } {\mathbb{R}}^{3}, \\ \int_{{\mathbb{R}}^3}u^2 \, \mathrm{d}x = a, \, \, u > 0, u \in H^{1}(\mathbb{R}^{3}), \end{array} \right. \end{equation*} $ where $ \lambda $ is a Lagrange multiplier, $ V(x) $ is a real-valued potential, $ a\in {\mathbb{R}}_{+} $ is a constant, $ p_{ \varepsilon} = \frac{10}{3} \pm \varepsilon $ and $ \varepsilon>0 $ is a sma...
