2011/08/29 by Francesca De Marchis, De Marchis, Francesca, Andrea Malchiodi +3
Computer Science · Mathematics · #35A01 #35A15 #35B33 #35B44 #35K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #math.FA #msc:35A01 #msc:35A15 #msc:35B33 #msc:35B44 #msc:35K55
paper · pdf · doi:10.48550/arxiv.1108.5576
The paper has been withdrawn. We need to fix an error
openalex publication_date 2011/08/29 · arxiv created 2011/09/17 · arxiv updated 2011/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
On a smooth bounded 2-dimensional domain Ω we study the heat flow ut=Δu +λ(t)ueu2 (λ(t) is such that d/dt ||u(t,⋅)||H10=0) introduced by T. Lamm, F. Robert and M. Struwe to investigate the Moser-Trudinger functional E(v)=∫Ω (ev2-1)dx, v∈ H10(Ω). We prove that if u blows-up as t→∞ and if E(u(t,⋅)) remains bounded, then for a sequence tk→∞ we have u(tk,⋅)\rightharpoonup 0 in H10 and ‖u(tk,⋅)‖H102→ 4πL for an integer L≥ 1. We couple these results with a topological technique to prove that if Ω is not contractible, then for every 0<Λ∈ ℝ ∖ 4 πℕ the functional E constrained to MΛ=\v∈ H10(Ω):||v||H102=Λ\ has a positive critical point. We prove that when Ω is the unit ball and Λ is large enough, then E|MΛ has no positive critical points, hence showing that the topological assumption on Ω is natural.