Normalized Solutions for nonlinear Schrödinger-Poisson equations involving nearly mass-critical exponents

We study the Schrödinger-Poisson-Slater equation \begin{equation*}\left{\begin{array}{lll} -Δu + λu + \big(|x|^{-1} \ast |u|^{2}\big)u = V(x) u^{ p_{\varepsilon}-1 }, , $\text{ in }$ $\mathbb{R}$^{3},\[2mm] \int_{$\mathbb{R}$^3}u^2 ,dx= 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, $p_{\epsilon }=\frac{10}{3}\pm \epsilon$ and $\epsilon >0$ is a small parameter. In this paper, we prove that it is the positive critical value of the potential $V$ that affects the existence of single-peak solutions for this problem. Furthermore, we prove the local uniqueness of the solutions we construct.