Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data

We consider the elliptic equation -Δu+u=0 in a bounded, smooth domain Ω in R2 subject to the nonlinear Neumann boundary condition ∂u∂ν=ɛeu. Here ɛ>0 is a small parameter. We prove that any family of solutions uɛ for which ɛ∫∂Ωeu is bounded, develops up to subsequences a finite number m of peaks ξi∈∂Ω, in the sense that ɛeu⇀2π∑k=1mδξi as ɛ→0. Reciprocally, we establish that at least two such families indeed exist for any given m⩾1.