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

Abstract

We construct multiple solutions to the nonlocal Liouville equation \begin{equation} \label{eqk} \tag{L} (-Δ)^{\frac{1}{2}} u = K(x) e^u \quad \mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form $K(x)=1+\epsilon \kappa (x)$ with $\epsilon \in (0,1)$ 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.

Accepted for publication on SIAM J. Math. Anal