Multiple solutions to the nonlocal Liouville equation in \(\mathbb{R}\)

Differential equations seminar

Speaker: Antonio J. Fernández (Universidad Autónoma de Madrid)

Abstract: We contruct multiple solutions to the Liouville type equation (\(-\Delta)^{\frac12}\) \(u = k(x) e^u\), in \(\mathbb{\mathbb{R}}\). 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 the existence of multiple solutions to the above equation bifurcating from the so-called Aubin-Talenti bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature \(k(x)\) on its boundary. Moreover, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS. The talk is based on joint works with L. Battaglia (Roma), M. Cozzi (Milano) and A. Pistoia (Roma).