Sign-changing solutions for the sinh-Poisson equation with Robin Boundary condition

2023/01/09 by Pablo Figueroa, Figueroa, Pablo, Leonelo Iturriaga +3
Computer Science · Mathematics · #35B25 #35B38 #35J25 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.03688

openalex publication_date 2023/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given ε∈ (0,1) and λ> 1, we address the existence of solutions for the Sinh-Poisson equation with Robin boundary value condition \begincases Δu+ε² (e^u - e^-u)=0 amp;in Ω
(∂ u)/(∂ν)+λu=0 amp;on ∂Ω, \endcases where Ω⊂ℝ² is a bounded smooth domain. We prove two existence results under a suitable relation between ε small and λ large. When Ω is symmetric with respect to an axis, we prove the existence of a family of solutions u_ε,λ concentrating at two points with different spin, both located on the symmetry line and close to the boundary. In the second result, we assume Ω is not simply connected and we construct sign-changing solutions concentrating at points located close to the boundary, each of them on a different connected component of the boundary.

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