Acta Mathematica

Uniqueness of non-linear ground states for fractional Laplacians in R

Rupert L. Frank and Enno Lenzmann

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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.

Article information

Source
Acta Math., Volume 210, Number 2 (2013), 261-318.

Dates
Received: 24 May 2011
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892706

Digital Object Identifier
doi:10.1007/s11511-013-0095-9

Mathematical Reviews number (MathSciNet)
MR3070568

Zentralblatt MATH identifier
1307.35315

Rights
2013 © Institut Mittag-Leffler

Citation

Frank, Rupert L.; Lenzmann, Enno. Uniqueness of non-linear ground states for fractional Laplacians in ${\mathbb{R}}$. Acta Math. 210 (2013), no. 2, 261--318. doi:10.1007/s11511-013-0095-9. https://projecteuclid.org/euclid.acta/1485892706


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