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

Abstract

We prove uniqueness of ground state solutions $Q=Q(|x|)\ge 0$ 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 $s\ge 1/2$. 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 $\ker L_{+}=\mathrm{span}\{Q^{\prime }\}$. 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.

45 pages

References in corpus (3)

• Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations
• Uniqueness of radial solutions for the fractional Laplacian
• Fractional Laplacian in Conformal Geometry

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