Existence and qualitative properties of concentrating solutions for the sinh-Poisson equation

Abstract

We prove the existence of nodal solutions for – Δu = ρ sinh u with Dirichlet boundary conditions in bounded, 2D, smooth and non-smooth domains. Indeed, for ρ positive and small enough, we show that there exist at least two pairs of solutions, which change sign exactly once, whose nodal lines intersect the boundary.