Asymptotic analysis for the Lane-Emden problem in dimension two

TL;DR

This paper surveys recent asymptotic results for solutions of the Lane-Emden problem in two dimensions as the exponent p approaches infinity, highlighting the behavior of solutions in smooth bounded domains.

Contribution

It provides a comprehensive overview of recent findings on the asymptotic behavior of solutions to the Lane-Emden problem in 2D as p tends to infinity.

Findings

Solutions exhibit concentration phenomena as p increases

Asymptotic profiles of solutions are characterized in the limit

The behavior depends on the geometry of the domain

Abstract

We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }\Omega u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array}\right. \end{equation} when $p>1$ and $\Omega\subset\mathbb R^2$ is a smooth bounded domain. The aim of the paper is to survey some recent results on the asymptotic behavior of solutions of (1) as the exponent $p\rightarrow \infty $.