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

TL;DR

This paper analyzes the asymptotic behavior of positive solutions to the Lane-Emden problem in two dimensions as the exponent tends to infinity, demonstrating energy quantization and convergence of the solution norm.

Contribution

It completes the asymptotic analysis for the problem in 2D, proving energy quantization and confirming a previous conjecture about solution behavior.

Findings

Energy quantization to multiples of 8πe

Convergence of the L-infinity norm to √e

Confirmation of the conjecture from prior work

Abstract

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].