NONUNIQUENESS FOR THE NONLOCAL LIOUVILLE EQUATION IN R AND APPLICATIONS

We construct multiple solutions to the nonlocal Liouville equation (-Delta)(1/2)u = K(x)e(u) in R . More precisely, for K of the form K(x) = 1 + epsilon kappa(x) with epsilon is an element of (0,1) small and kappa is an element of C-1,C-alpha(R) boolean AND L-infinity(R) for some alpha > 0, we prove the existence of multiple solutions to the above equation 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 & ouml;dinger equation.