Uniqueness of positive bound states with multi-bump for nonlinear Schr\"odinger equations

TL;DR

This paper proves the uniqueness of positive multi-bump solutions for a class of nonlinear Schrödinger equations, focusing on solutions concentrating at multiple critical points of the potential, even allowing degeneracy.

Contribution

It establishes the uniqueness of multi-bump solutions concentrating at multiple critical points of the potential for small epsilon, including degenerate critical points.

Findings

Uniqueness of positive multi-bump solutions under specified conditions.

Solutions concentrate at multiple critical points of the potential.

Degeneracy of critical points is permitted in the analysis.

Abstract

We are concerned with the following nonlinear Schr\"odinger equation $$-\varepsilon^2\Delta u+ V(x)u=|u|^{p-2}u,~u\in H^1(\R^N),$$ where $N\geq 3$, $2<p<\frac{2N}{N-2}$. For $\varepsilon$ small enough and a class of $V(x)$, we show the uniqueness of positive multi-bump solutions concentrating at $k$ different critical points of $V(x)$ under certain assumptions on asymptotic behavior of $V(x)$ and its first derivatives near those points. The degeneracy of critical points is allowed in this paper.