Uniqueness for the nonlocal Liouville equation in R

We prove uniqueness of solutions for the nonlocal Liouville equation(−Δ)1/2w=Kewin R with finite total Q-curvature ∫RKewdx0 is assumed to be a positive, symmetric-decreasing function satisfying suitable regularity and decay bounds. In particular, we obtain uniqueness of solutions in the Gaussian cas...

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Bibliographic Details
Published in:Journal of functional analysis 2022-12, Vol.283 (12), p.109712, Article 109712
Main Authors: Ahrend, Maria, Lenzmann, Enno
Format: Article
Language:English
Subjects:
Fractional Laplacian
Liouville equation
Nonlocal Q-curvature
Uniqueness
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Summary:We prove uniqueness of solutions for the nonlocal Liouville equation(−Δ)1/2w=Kewin R with finite total Q-curvature ∫RKewdx0 is assumed to be a positive, symmetric-decreasing function satisfying suitable regularity and decay bounds. In particular, we obtain uniqueness of solutions in the Gaussian case with K(x)=exp⁡(−x2). Our uniqueness proof exploits a connection of the nonlocal Liouville equation to ground state solitons for Calogero–Moser derivative NLS, which is a completely integrable PDE recently studied by P. Gérard and the second author.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2022.109712