On Coron's problem for weakly coupled elliptic systems

2016/10/25 by Angela Pistoia, Nicola Soave, Pistoia, Angela +1 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1610.07762

arxiv created 2016/10/25 · openalex publication_date 2016/10/25 · arxiv updated 2016/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the following critical weakly coupled elliptic system \begincases -Δu_i = μ_i |u_i|^{2^*-2}u_i + ∑_{j ≠ i} β_ij |u_j|^{(2^*)/(2)} |u_i|^{(2^*-4)/(2)} u_i in Ω_ε u_i >0 in Ω_ε u_i = 0 on ∂ Ω_ε,\endcases i =1,…,m, in a domain Ω_ε ⊂ ℝ^N, N=3,4, with small shrinking holes as the parameter ε → 0. We prove the existence of positive solutions of two different types: either each density concentrates around a different hole, or we have groups of components such that all the components within a single group concentrate around the same point, and different groups concentrate around different points.

Citations

• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

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• Segregated solutions for nonlinear Schrödinger systems with sublinear coupling terms
• Segregated Bubbling Solutions for a Critical Schrödinger System of Brezis--Nirenberg Type with Sublinear Competitive Coupling
• On least energy solutions for a nonlinear Schrödinger system with K-wise interaction

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