Sharp boundary concentration for a two-dimensional nonlinear Neumann problem

TL;DR

This paper investigates the boundary concentration phenomena of positive solutions to a nonlinear Neumann problem in two dimensions as the nonlinearity exponent grows large, revealing sharp asymptotic profiles and energy quantization.

Contribution

It provides a detailed analysis of boundary concentration, energy quantization, and asymptotic profiles for solutions as the exponent tends to infinity.

Findings

Energy quantization at boundary points

Sharp convergence of solutions in $L^{}$ norm

Characterization of local asymptotic profiles

Abstract

We consider the elliptic equation $-\Delta u+ u=0$ in a bounded, smooth domain $\Omega\subset\mathbb R^{2}$ subject to the nonlinear Neumann boundary condition $\partial u/\partial\nu = |u|^{p-1}u$ on $\partial\Omega$ and study the asymptotic behavior as the exponent $p\rightarrow +\infty$ of families of positive solutions $u_p$ satisfying uniform energy bounds. We prove energy quantization and characterize 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 $L^{\infty}$-norm.