Coron's problem for the critical Lane-Emden system

TL;DR

This paper proves the existence of positive solutions concentrating around a small hole in a bounded domain for the critical Lane-Emden system, a first such result in this setting, contrasting with known non-existence on star-shaped domains.

Contribution

It establishes the first existence results for the critical Lane-Emden system on bounded domains with small holes, detailing solution concentration behavior as the hole shrinks.

Findings

Positive solutions concentrate near the hole as its radius approaches zero.

First existence proof for the critical Lane-Emden system on bounded domains with small holes.

Contrasts with known non-existence results on star-shaped domains.

Abstract

In this paper, we address the solvability of the critical Lane-Emden system \[\begin{cases} -\Delta u=|v|^{p-1}v &\mbox{in } \Omega_\epsilon,\\ -\Delta v=|u|^{q-1}u &\mbox{in } \Omega_\epsilon,\\ u=v=0 &\mbox{on } \partial \Omega_\epsilon, \end{cases}\] where $N \ge 4$, $p \in (1,\frac{N-1}{N-2})$, $\frac{1}{p+1} + \frac{1}{q+1}=\frac{N-2}{N}$, and $\Omega_\epsilon$ is a smooth bounded domain with a small hole of radius $\epsilon > 0$. We prove that the system admits a family of positive solutions that concentrate around the center of the hole as $\epsilon \to 0$, obtaining a concrete qualitative description of the solutions as well. To the best of our knowledge, this is the first existence result for the critical Lane-Emden system on a bounded domain, while the non-existence result on star-shaped bounded domains has been known since the early 1990s due to Mitidieri (1993) [30] and van der Vorst (1991) [36].