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
Nonuniqueness for the Nonlocal Liouville Equation in R and Applications / L. Battaglia, M. Cozzi, A.J. Fernández, A. Pistoia. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 55:5(2023 Sep 26), pp. 4816-4842. [10.1137/22M1538004]
Nonuniqueness for the Nonlocal Liouville Equation in R and Applications
M. CozziSecondo
;
2023
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 equationFile | Dimensione | Formato | |
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