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

Abstract

We consider the elliptic equation -Delta u+u=O in a bounded, smooth domain ohm in R-2 subject to the nonlinear Neumann boundary condition delta u/delta v = epsilon e(u). Here epsilon > 0 is a small parameter. We prove that any family of solutions u(epsilon) for which epsilon integral(partial derivative ohm)e(u) is bounded, develops up to subsequences a finite number m of peaks xi(i) is an element of partial derivative ohm, in the sense that epsilon e(u) -> 2 pi Sigma(m)(k=1) delta(zeta i) as epsilon -> 0. Reciprocally, we establish that at least two such families indeed exist for any given m >= 1.