Convergence for a planar elliptic problem with large exponent Neumann data

Abstract

We study positive solutions $u_p$ of the nonlinear Neumann elliptic problem $\Delta u =u$ in $\Omega $, $\partial u/\partial\nu = |u|^{p-1}u$ on $\partial\Omega$, where $\Omega $ is a bounded open smooth domain in $\mathbb{R}^2$. We investigate the asymptotic behavior of families of solutions $u_p$ satisfying an energy bound condition when the exponent $p$ is getting large. Inspired by the work of Davila-del Pino-Musso \cite{DavilaDM}, we prove that $u_p$ is developing $m$ peaks $x_i\in\partial \Omega$, in the sense $u_p^p/\int_{\partial \Omega}u_p^p$ approaches the sum of $m$ Dirac masses at the boundary and we determine the localization of these concentration points.