Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$

Rupert L. Frank Department of Mathematics, Princeton University Enno Lenzmann Institute for Mathematical Sciences, University of Copenhagen

Mathematical Physics Spectral Theory and Operator Algebra mathscidoc:1701.22001

Acta Mathematica, 210, (2), 261-318, 2011.5
We prove uniqueness of ground state solutions$Q$=$Q$(|$x$|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$ where 0 <$s$< 1 and 0 <$α$< 4$s$/(1−2$s$) for $${s<\frac{1}{2}}$$ and 0 <$α$<$∞$for $${s\geq \frac{1}{2}}$$ . 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 $${s=\frac{1}{2}}$$ and$α$= 1 in [5] for the Benjamin–Ono equation.
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@inproceedings{rupert2011uniqueness,
  title={Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$ },
  author={Rupert L. Frank, and Enno Lenzmann},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203400823704771},
  booktitle={Acta Mathematica},
  volume={210},
  number={2},
  pages={261-318},
  year={2011},
}
Rupert L. Frank, and Enno Lenzmann. Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$ . 2011. Vol. 210. In Acta Mathematica. pp.261-318. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203400823704771.
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