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.

arXiv admin note: text overlap with arXiv:1602.06919 by other authors