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

Nonuniqueness for the Nonlocal Liouville Equation in \(\mathbb{R}\) and Applications / Battaglia, Luca; Cozzi, Matteo; Fernández, Antonio J.; Pistoia, Angela. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 55:5(2023), pp. 4816-4842. [10.1137/22m1538004]

Nonuniqueness for the Nonlocal Liouville Equation in \(\mathbb{R}\) and Applications

Cozzi, Matteo;Pistoia, Angela
2023

Abstract

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
2023
Liouville equation, half-Laplacian, multiplicity results, Lyapunov-Schmidt reduction, Brouwer degree
01 Pubblicazione su rivista::01a Articolo in rivista
Nonuniqueness for the Nonlocal Liouville Equation in \(\mathbb{R}\) and Applications / Battaglia, Luca; Cozzi, Matteo; Fernández, Antonio J.; Pistoia, Angela. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 55:5(2023), pp. 4816-4842. [10.1137/22m1538004]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1705194
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