Nonuniqueness for the Nonlocal Liouville Equation in R and Applications

We construct multiple solutions to the nonlocal Liouville equation ( Δ ) 1 2 u = K(x)eu in R. More precisely, for K of the form K(x) = 1 + ϵ \kappa (x) with ϵ ∈ (0, 1) small and κ ∈ C1,α (R)∩ L∞ (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\"odinger equation