Convergence for a planar elliptic problem with large exponent Neumann data

TL;DR

This paper analyzes the asymptotic behavior of positive solutions to a nonlinear elliptic problem with large exponent Neumann boundary data, revealing the development of boundary peaks and their localization.

Contribution

It proves that solutions develop multiple boundary peaks as the exponent grows large, extending previous work and identifying the concentration points on the boundary.

Findings

Solutions form multiple boundary peaks as p increases.

The normalized solutions converge to a sum of Dirac masses at boundary points.

Localization of concentration points on the boundary is explicitly determined.

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.