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.
Battaglia, L., Cozzi, M., Fernandez, A.J., Pistoia, A. (2023). NONUNIQUENESS FOR THE NONLOCAL LIOUVILLE EQUATION IN R AND APPLICATIONS. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 55(5), 4816-4842 [10.1137/22M1538004].
NONUNIQUENESS FOR THE NONLOCAL LIOUVILLE EQUATION IN R AND APPLICATIONS
Battaglia L.;Fernandez A. J.;
2023-01-01
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
