Asymptotic analysis and energy quantization for the Lane-Emden problem in dimension two

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].