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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 gettinglarge. Inspired by the work of Davila-del Pino-Musso \cite{DavilaDM}, we provethat $u_p$ 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 atthe boundary and we determine the localization of these concentration points.

Convergence for a planar elliptic problem with large exponent Neumann data