Uniqueness for the Nonlocal Liouville Equation in ℝ

2022/03/29 by Ahrend, Maria, Lenzmann, Enno
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences

paper · doi:10.48550/arxiv.2203.15843

Abstract

We prove uniqueness of solutions for the nonlocal Liouville equation (-Δ)^1/2 w = K e^w in ℝ with finite total Q-curvature ∫_ℝ K e^w dx< +∞. Here the prescribed Q-curvature function K=K(|x|) > 0 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.

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