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

Given $ε\in (0,1)$ and $λ> 1$, we address the existence of solutions for the Sinh-Poisson equation with Robin boundary value condition $$ \begin{cases} Δu+ε^2 (e^{u} - e^{-u})=0 &\mbox{in }Ω\\ \frac{\partial u}{\partialν}+λu=0 &\mbox{on }\partialΩ, \end{cases} $$ where $Ω\subset\mathbb{R}^2$ 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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