Coron's problem for the critical Lane-Emden system

2022/10/24 by Jin, Sangdon, Kim, Seunghyeok
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.13068

Abstract

In this paper, we address the solvability of the critical Lane-Emden system \begincases -Δu=|v|^p-1v amp;in Ω_ε,
-Δv=|u|^q-1u amp;in Ω_ε,
u=v=0 amp;on ∂ Ω_ε, \endcases where N ≥ 4, p ∈ (1,(N-1)/(N-2)), (1)/(p+1) + (1)/(q+1)=(N-2)/(N), and Ω_ε is a smooth bounded domain with a small 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 (1991) [36].

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