Mixed interior and boundary nodal bubbling solutions for a sinh-Poisson equation

We consider here the semilinear equation Δu +2ε² sinh u = 0 posed on a bounded smooth domain Ω in R{double-struck}² with homogeneous Neumann boundary condition, where ε > 0 is a small parameter. We show that for any given nonnegative integers k and l with k+l ≥ 1, there exists a family of solutions u_ε that develops 2k interior and 2l boundary singularities for ε sufficiently small, with the property that where (ζ₁,...,ζ_2(k+l)) are critical points of some functional defined explicitly in terms of the associated Green function.