Non-uniqueness for the nonlocal Liouville equation in $\mathbb{R}$ and applications

TL;DR

This paper constructs multiple solutions to a nonlocal Liouville equation in one dimension, demonstrating non-uniqueness and applications to geometric problems and soliton solutions in integrable systems.

Contribution

It introduces a method to find multiple solutions to the nonlocal Liouville equation with prescribed curvature, revealing non-uniqueness and linking to geometric and physical applications.

Findings

Existence of multiple solutions bifurcating from bubbles.

Solutions correspond to flat metrics with prescribed boundary curvature.

Implications for multiple ground state solitons in Calogero-Moser NLS.

Abstract

We construct multiple solutions to the nonlocal Liouville equation \begin{equation} \label{eqk} \tag{L} (-\Delta)^{\frac{1}{2}} u = K(x) e^u \quad \mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form $K(x) = 1+\varepsilon \kappa(x)$ with $\varepsilon \in (0,1)$ small and $\kappa \in C^{1,\alpha}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ for some $\alpha > 0$, we prove existence of multiple solutions to \eqref{eqk} bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS.