Coron's problem for the critical Lane-Emden system

We address the solvability of the critical Lane-Emden system in a smooth bounded domain with a small spherical hole of radius ϵ>0. We prove that the system admits a family of positive solutions that concentrate around the center of the hole as ϵ→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 (1992) [36].