Uniqueness for the Nonlocal Liouville Equation in $\mathbb{R}$

Abstract

We prove uniqueness of solutions for the nonlocal Liouville equation $$ (-Δ)^{1/2} w = K e^w \quad \mbox{in $\mathbb{R}$} $$ with finite total $Q$-curvature ${\int }_{\mathbb{R}}K{e}^{w}dx<+\infty$. 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 $K(x)=\exp (-x^2)$. 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.

18 pages. Revised version with short Appendix B on relation to the monotonicity formula. Some typos fixed and few remarks added. Comments are welcome