Removable singularity of the polyharmonic equation

2007/02/07 by Shu-Yu Hsu, Hsu, Shu-Yu
Computer Science · Mathematics · #35B65 #35J30 #35J99 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.math/0702171

openalex publication_date 2007/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let x₀∈Ω⊂ℝⁿ, n≥ 2, be a domain and let m≥ 2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω∖\x₀\ has a removable singularity at x₀ if and only if |Δ^ku(x)|=o(|x-x₀|^2-n) ∀ k=0,1,2,...,m-1 as |x-x₀|→ 0 for n≥ 3 and =o(log (|x-x₀|^-1)) ∀ k=0,1,2,...,m-1 as |x-x₀|→ 0 for n=2. For m≥ 2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x-x₀|^2m-n) as |x-x₀|→ 0 for n≥ 3 and |u(x)| =o(|x-x₀|^2m-2log (|x-x₀|^-1)) as |x-x₀|→ 0 for n=2.

Citations

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

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