Convergence for a planar elliptic problem with large exponent Neumann data

Abstract

We study positive solutions up of the nonlinear Neumann elliptic problem Δu=u in Ω, ∂u/∂ν=|u|p−1u on ∂Ω, where Ω is a bounded open smooth domain in R2. We investigate the asymptotic behavior of families of solutions up satisfying an energy bound condition when the exponent p is getting large. Inspired by the work of Davila-del Pino-Musso [8], we prove that up is developing m peaks xi∈∂Ω, in the sense upp/∫∂Ωupp approaches the sum of m Dirac masses at the boundary and we determine the localization of these concentration points.