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2013 Uniqueness of non-linear ground states for fractional Laplacians in R
Rupert L. Frank, Enno Lenzmann
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Acta Math. 210(2): 261-318 (2013). DOI: 10.1007/s11511-013-0095-9

Abstract

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation (Δ)sQ+QQα+1=0inR,where 0 < s < 1 and 0 < α < 4s/(1−2s) for s<12 and 0 < α <  for s12. Here (−Δ)s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s=12 and α = 1 in [5] for the Benjamin–Ono equation.

As a technical key result in this paper, we show that the associated linearized operator L+ = (−Δ)s+1−(α+1)Qα is non-degenerate; i.e., its kernel satisfies ker L+ = 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 non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.

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Rupert L. Frank. Enno Lenzmann. "Uniqueness of non-linear ground states for fractional Laplacians in R." Acta Math. 210 (2) 261 - 318, 2013. https://doi.org/10.1007/s11511-013-0095-9

Information

Received: 24 May 2011; Published: 2013
First available in Project Euclid: 31 January 2017

zbMATH: 1307.35315
MathSciNet: MR3070568
Digital Object Identifier: 10.1007/s11511-013-0095-9

Rights: 2013 © Institut Mittag-Leffler

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Vol.210 • No. 2 • 2013
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