Abstract: 
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 socalled AubinTalenti bubbles. These solutions provide examples of flat metrics in the halfplane with prescribed geodesic curvature $k(x)$ on its boundary. Moreover, they imply the existence of multiple ground state soliton solutions for the CalogeroMoser derivative NLS. The talk is based on joint works with L. Battaglia (Roma), M. Cozzi (Milano) and A. Pistoia (Roma). 
