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

Abstract

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 $\hat{I}»$ 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.