Non-uniqueness for the nonlocal Liouville equation in ℝ and applications

2022/11/22 by Battaglia, Luca, Cozzi, Matteo, Fernández, Antonio J. +1
#30F45 #35A02 #Analysis of PDEs (math.AP) #FOS: Mathematics #primary 35R11 #secondary 35C08

paper · doi:10.48550/arxiv.2211.12106

Abstract

We construct multiple solutions to the nonlocal Liouville equation (-Δ)^(1)/(2) u = K(x) e^u in ℝ. More precisely, for K of the form K(x) = 1+ε κ(x) with ε ∈ (0,1) small and κ∈ C^1,α(ℝ) ∩ L^∞(ℝ) for some α> 0, we prove existence of multiple solutions to \eqrefeqk 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 NLS.

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