Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola

TL;DR

This paper investigates the asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola, extending previous results to all cases and general domains.

Contribution

It generalizes the analysis of least energy solutions to the Lane-Emden system to include cases with p<1 and arbitrary smooth bounded domains.

Findings

Extended asymptotic analysis to all p>0 cases.

Included non-convex domains in the analysis.

Provided comprehensive understanding near the critical hyperbola.

Abstract

The Lane-Emden system is written as \begin{equation*} \begin{cases} -\Delta u = v^p &\text{in } \Omega,\\ -\Delta v = u^q &\text{in } \Omega,\\ u, v > 0 &\text{in } \Omega,\\ u = v = 0 &\text{on } \partial \Omega \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in the Euclidean space $\mathbb{R}^n$ for $n \ge 3$ and $0< p< q <\infty$. The asymptotic behavior of least energy solutions near the critical hyperbola was studied by Guerra \cite{G} when $p \geq 1$ and the domain is convex. In this paper, we cover all the remaining cases $p < 1$ and extend the results to any smooth bounded domain.