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

Abstract

We complete the study of the asymptotic behavior, as $p\to +\infty$, of the positive solutions to \[ \left\{\begin{array}{lr}-Δu= u^p & \mbox{in}Ω\\ u=0 &\mbox{on}\partial Ω\end{array}\right. \] 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].