Asymptotic profile of positive solutions of Lane-Emden problems 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 p approaches infinity, under a specific energy condition, providing a comprehensive description of their profile.

Contribution

It offers a complete characterization of the asymptotic profile of solutions to the Lane-Emden problem in 2D as p tends to infinity, under an energy convergence condition.

Findings

Detailed asymptotic profile of solutions as p→∞

Conditions under which the asymptotic behavior is characterized

Insights into the energy distribution of solutions

Abstract

We consider families $u_p$ of solutions to the problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= u^p & \mbox{ in }\Omega\\ u>0 & \mbox{ in }\Omega\\ u=0 & \mbox{ on }\partial \Omega \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $\Omega$ is a smooth bounded domain of $\mathbb R^2$. We give a complete description of the asymptotic behavior of $u_p$ as $p\rightarrow +\infty$, under the condition \[p\int_{\Omega} |\nabla u_p|^2\,dx\rightarrow \beta\in\mathbb R\qquad\mbox{ as $p\rightarrow +\infty$}.\]