Special Session 47: Singular limits in elliptic and parabolic PDEs

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

Antonio J. Fern\`andez
Universidad Aut\`onoma de Madrid
Co-Author(s):    Luca Battaglia, Matteo Cozzi and Angela Pistoia
We construct multiple solutions to the Liouville type equation $$ (-\Delta)^{\frac12} u = k(x) e^u, \quad \textup{ in } \mathbb{R} $$ More precisely, for $k$ of the form $k(x) = 1+\epsilon\kappa(x)$ with $\epsilon \in (0,1)$ small and $\kappa \in C^{1,\alpha}(\R) \cap L^{\infty}(\mathbb{\R})$ 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).