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Published June 2013 | public
Journal Article

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

Abstract

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation (−Δ)^sQ+Q−Q^(α+1)=0inR, ( − Δ ) s Q + Q − Q α + 1 = 0 i n R , where 0 < s < 1 and 0 < α < 4s/(1−2s) for s<12 s < 1 2 and 0 < α < ∞ for s≥12 s ≥ 1 2 . 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 s = 1 2 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)Q^α 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.

Additional Information

© 2013 Institut Mittag-Leffler. Received May 24, 2011. R. F. acknowledges support from NSF grant PHY-0652854. E. L. was supported by a Steno fellowship from the Danish science research council, and he also gratefully acknowledges partial support from NSF grant DMS-0702492.

Additional details

Created:
August 19, 2023
Modified:
October 24, 2023