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

TL;DR

This paper proves the existence of sign-changing solutions for the sinh-Poisson equation with Robin boundary conditions, showing solutions concentrate near the boundary under certain parameter conditions and domain symmetries.

Contribution

It introduces new existence results for solutions with specific concentration behaviors in symmetric and non-simply connected domains.

Findings

Solutions concentrate at boundary points with different spins in symmetric domains.

Sign-changing solutions concentrate near boundary components in non-simply connected domains.

Existence is established for small epsilon and large lambda under domain symmetry assumptions.

Abstract

Given $\epsilon \in (0,1)$ and $\lambda > 1$, we address the existence of solutions for the Sinh-Poisson equation with Robin boundary value condition $$ \begin{cases} \Delta u+\epsilon^2 (e^{u} - e^{-u})=0 &\mbox{in }\Omega\\ \frac{\partial u}{\partial\nu}+\lambda u=0 &\mbox{on }\partial\Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^2$ is a bounded smooth domain. We prove two existence results under a suitable relation between $\epsilon$ small and $\lambda$ large. When $\Omega$ is symmetric with respect to an axis, we prove the existence of a family of solutions $u_{\epsilon,\lambda}$ concentrating at two points with different spin, both located on the symmetry line and close to the boundary. In the second result, we assume $\Omega$ 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.