Nonuniqueness for the nonlocal Liouville equation in \(\mathbb{R}\) and applications

We construct multiple solutions to the nonlocal Liouville equation ( - \Delta )1 2u = K(x)eu in \BbbR . More precisely, for K of the form K(x) = 1 + \varepsilon \kappa (x) with \varepsilon \in (0,1) small and \ k a p p a \in C1,\alpha (\BbbR )\cap L\infty (\BbbR ) for some \alpha >0, we prove the existence of multiple solutions to the above equa tion 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 nonlinear Schr\"odinger equation