On the supercritical mean field equation on pierced domains

2013/12/13 by Mohameden Ould Ahmedou, Angela Pistoia, Ahmedou, Mohameden Ould +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1312.3768

arxiv created 2013/12/13 · openalex publication_date 2013/12/13 · arxiv updated 2013/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem (P_\eps) Δu +λe^u\over∫_{Ω∖ B(ξ,\eps)}e^u=0 \hboxin Ω∖ B(ξ,\eps), u =0 \hboxon ∂\(Ω∖ B(ξ,\eps)\), where Ω is a smooth bounded open domain in \rr² which contains the point ξ. We prove that if λ>8π, problem (P_\eps) has a solutions u_\eps such that u_\eps(x)→ 8π+ λ\over2 G(x,ξ) \hboxuniformly on compact sets of Ω∖\ξ\ as \eps goes to zero. Here G denotes Green's function of Dirichlet Laplacian in Ω. If λ\not∈ 8π\mathbb N we will not make any symmetry assumptions on Ω, while if λ∈ 8π\mathbb N we will assume that Ω is invariant under a rotation through an angle 8π²\over \la around the point ξ.

Citations

• Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter

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