On the regularity and partial regularity of extremal solutions of a Lane-Emden system

2016/11/16 by Hatem Hajlaoui, Hajlaoui, Hatem
Mathematics · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Meromorphic and Entire Functions #Nonlinear Differential Equations Analysis #math.AP

paper · pdf · doi:10.48550/arxiv.1611.05488

arxiv created 2016/11/16 · openalex publication_date 2016/11/16 · arxiv updated 2016/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the system -Δu =λ(v+1)^p, -Δv = γ(u+1)^θ on a smooth bounded domain Ω in ℝ^N with the Dirichlet boundary condition u=v=0 on ∂ Ω. Here λ,γ are positive parameters. Let x₀ be the largest root of the polynomial H(x) = x⁴ - (16pθ(p+1)(θ+1))/((pθ-1)²)x² + (16pθ(p+1)(θ+1)(p+θ+2))/((pθ-1)³)x -(16pθ(p+1)²(θ+1)²)/((pθ-1)⁴). We show that the extremal solutions associated to the above system are bounded provided N<2+2x₀. This improves the previous work in \citeco1. We also prove that, if N≥ 2+2x₀, then the singular set of any extremal solution has Hausdorff dimension less or equal to N-(2+2x₀).

Related

• Regularity of stable solutions of a Lane-Emden type system
• Regularity of the extremal solutions associated to elliptic systems
• Regularity of the extremal solutions associated to elliptic systems
• The Lane-Emden system near the critical hyperbola on nonconvex domains
• Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions