Uniqueness for the nonlocal Liouville equation in R

We prove uniqueness of solutions for the nonlocal Liouville equation (-Delta)1/2w = Kew in R with finite total Q-curvature acute accent R Kew dx < +infinity. 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. Gerard and the second author.(c) 2022 Published by Elsevier Inc.