Sharp boundary concentration for a two-dimensional nonlinear Neumann problem

Abstract

We consider the elliptic equation $-\hat{I}”u+u=0$ in a bounded, smooth domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ subject to the nonlinear Neumann boundary condition $\partial u/\partial \hat{I}½=|u{|}^{p-1}u$ on $\partial \hat{I}\text{c◯}$ and study the asymptotic behavior as the exponent $p\to +\infty$ 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.