Normalized Solutions for nonlinear Schr\"{o}dinger-Poisson equations involving nearly mass-critical exponents

TL;DR

This paper investigates the existence and uniqueness of positive solutions to a nonlinear Schr"{o}dinger-Poisson equation with nearly mass-critical exponents, highlighting the critical role of the potential's positive critical value.

Contribution

It establishes the influence of the potential's critical value on solution existence and proves local uniqueness for solutions in the nearly mass-critical regime.

Findings

Existence of single-peak solutions depends on the potential's critical value.

Positive critical value of potential determines solution existence.

Solutions are locally unique near the constructed solutions.

Abstract

We study the Schr\"{o}dinger-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},\\[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_{\varepsilon} = \frac{10}{3} \pm \varepsilon$ and $\varepsilon>0$ 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.