Qualitative analysis of multi-peak solutions for Nonlinear Schr\"{o}dinger equations with nearly critical Sobolev exponents

TL;DR

This paper investigates the qualitative properties, including local uniqueness and Morse index, of multi-peak solutions to a nearly critical nonlinear Schrödinger equation with potential functions having multiple non-degenerate critical points.

Contribution

It establishes the local uniqueness and Morse index of multi-peak solutions using blow-up analysis and Pohozaev identities, extending prior existence results.

Findings

Proves local uniqueness of multi-peak solutions.

Determines Morse index for these solutions.

Uses blow-up analysis based on Pohozaev identities.

Abstract

In this paper, we are concerned with qualitative properties of multi-peak solutions of the following nonlinear Schr\"{o}dinger equations \begin{equation*} -\Delta u+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.