  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.
