Coron's problem for the critical Lane-Emden system

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].