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

TL;DR

This work analyzes normalized spike-type solutions to a Schrödinger–Poisson–Slater system in $\mathbb{R}^3$ with exponents approaching the mass-critical value $\bar p=\frac{10}{3}$ under a mass constraint. The authors employ a Lyapunov–Schmidt finite-dimensional reduction to construct single-peak solutions concentrating at a nondegenerate critical point $b_0$ of a nontrivial potential $V$, with the Lagrange multiplier $\lambda_\varepsilon$ determined via a mass-crossing condition. They establish existence of normalized solutions for small $\varepsilon$ in two mass-regime cases and prove local uniqueness of these normalized spikes, including precise asymptotics and the role of the positive critical value $V_0=V(b_0)$. The results extend prior normalized-solution results to nontrivial potentials and clarify how concentration and multiplicity are governed by $V_0$ and the near mass-critical perturbation.

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 $λ$ 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.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.3
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5