Sharp boundary concentration for a two-dimensional nonlinear Neumann problem*

Abstract

Abstract We consider the elliptic equation − Δ u + u = 0 in a bounded, smooth domain Ω ⊂ R 2 subject to the nonlinear Neumann boundary condition ∂ u / ∂ ν = | u | p − 1 u on ∂ Ω and study the asymptotic behaviour as the exponent p → + ∞ of families of positive solutions up satisfying uniform energy bounds. We prove energy quantisation and characterise the boundary concentration. In particular we describe the local asymptotic profile of the solutions around each concentration point and get sharp convergence results for the L ∞ -norm.