Given ϵ ∈ (0, 1) and λ > 1, we address the existence of solutions for the sinh-Poisson equation with a Robin boundary value condition \(\begin{cases}\Delta u +\epsilon^{2}(e^{u}-e^{-u})=0 & \text{in}\ \mathbf{\Omega},\\{\partial u \over \partial \nu}+\lambda u=0 & \text{on}\ \partial \mathbf{\Omega},\end{cases}\) 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 uϵ,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 several points located close to the boundary, each of them on a different connected component of the boundary.
