Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q + Q - Q^{\alpha+1} = 0$ in $\mathbb{R}$

TL;DR

This paper proves the uniqueness and nondegeneracy of ground state solutions for a fractional Laplacian nonlinear equation in one dimension, extending previous results and providing key spectral properties relevant for stability and blowup analysis.

Contribution

It establishes the uniqueness and nondegeneracy of ground states for fractional Laplacian equations, using novel techniques that generalize prior specific cases.

Findings

Proved uniqueness of ground states for the fractional Laplacian equation.

Showed the associated linearized operator is nondegenerate.

Provided spectral properties crucial for stability analysis.

Abstract

We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-\Delta)^s Q + Q - Q^{\alpha+1}= 0$ in $\mathbb{R}$, where $0 < s < 1$ and $0 < \alpha < \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \alpha < \infty$ for $s \geq 1/2$. Here $(-\Delta)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $\alpha=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\Delta)^s + 1 - (\alpha+1) Q^\alpha$ is nondegenerate; i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span}\, \{Q'\}$. This result about $L_+$ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.