Coron's problem for the critical Lane-Emden system

In this paper, we address the solvability of the critical Lane-Emden system $$\{\begin{array}{ll}-\mathrm{\Delta }u=\mathrm{\mid }v{\mathrm{\mid }}^{p-1}v& in{\mathrm{\Omega }}_{\epsilon },\\ -\mathrm{\Delta }v=\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u& in{\mathrm{\Omega }}_{\epsilon },\\ u=v=0& on\mathrm{\partial }{\mathrm{\Omega }}_{\epsilon },\end{array}$$ 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 ${\mathrm{\Omega }}_{\epsilon }$ 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].