Qualitative analysis of multi-peak solutions for Nonlinear Schrödinger equations with nearly critical Sobolev exponents

TL;DR

This work advances the qualitative understanding of multi-peak solutions to the nonlinear Schrödinger equation with nearly critical Sobolev exponents in $\mathbb{R}^N$. Building on existing existence results, it employs blow-up analysis and local Pohozaev identities within a projected multi-peak framework to derive precise asymptotics for peak parameters and to prove local uniqueness of multi-peak states. It further characterizes the linearized spectrum to compute the Morse index and establish non-degeneracy, showing the Morse index equals the sum of the Hessian indices of the potential at the peak locations plus the number of peaks. Together, these results illuminate the stability and multiplicity structure of semi-classical states concentrated at stable critical points of the potential $V$.

Abstract

In this paper, we are concerned with qualitative properties of multi-peak solutions of the following nonlinear Schrödinger equations \begin{equation*} -Δu+V(x)u= u^{p-\varepsilon},\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb{R}^N, \end{equation*} where $V(x)$ is a nonnegative continuous function, $\varepsilon>0$, $p=\frac{N+2}{N-2}$, $N\geq6$. The existence of multi-peak solutions has been obtained by Cao et al. (Calc. Var. Partial Differential Equations, 64: 139, 2025). The main objective in this paper is to establish the local uniqueness and Morse index of the multi-peak solutions in \cite{CLl1} provided that $V(x)$ possesses $k$ non-degenerate critical points by using the blow-up analysis based on Pohozaev identities.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Remark 1.4
• Lemma 2.1
• proof : Proof.
• Lemma 2.2
• proof : Proof.
• Lemma 2.3
• proof : Proof.
• Proposition 2.4