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).