Supercritical problems in domains with thin toroidal holes

2013/06/01 by Seunghyeok Kim, Angela Pistoia, Kim, Seunghyeok +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1306.0099

arxiv created 2013/06/01 · openalex publication_date 2013/06/01 · arxiv updated 2013/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the Lane-Emden-Fowler equation (P)_ε \Δu+|u|^q-2u=0 \hboxin \mathcal D_ε, u=0 \hboxon ∂\mathcal D_ε. Here \mathcal D_ε= \mathcal D ∖ \x ∈ \mathcal D : dist(x,Γ_ℓ)≤ ε\, \mathcal D is a smooth bounded domain in ℝ^N, Γ_ℓ is an ℓ-dimensional closed manifold such that Γ_ℓ ⊂ \mathcal D with 1≤ ℓ ≤ N-3 and q=2(N-ℓ)\over N-ℓ-2. We prove that, under some symmetry assumptions, the number of sign changing solutions to (P)_ε increases as ε goes to zero.

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