  We complete the study of the asymptotic behavior, as $p\rightarrow +\infty$,
of the positive solutions to \[ \left\{\begin{array}{lr}-\Delta u= u^p &
\mbox{in}\Omega\\ u=0 &\mbox{on}\partial \Omega \end{array}\right. \] when
$\Omega$ is any smooth bounded domain in $\mathbb R^2$, started in [4]. In
particular we show quantization of the energy to multiples of $8\pi e$ and
prove convergence to $\sqrt{e}$ of the $L^{\infty}$-norm, thus confirming the
conjecture made in [4].
