2013/04/06 by Mónica Clapp, Clapp, Mónica, Jorge Faya +3
Computer Science · Mathematics · #2010: 35J60 #35J20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1304.1907
openalex publication_date 2013/04/06 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We consider the supercritical problem -Δv=|v|p-2v in Θε, v=0 on ∂Θε, where Θ is a bounded smooth domain in ℝN, N≥3, p>2∗:=2N/(N-2), and Θε is obtained by deleting the ε-neighborhood of some sphere which is embedded in Θ. In some particular situations we show that, for ε>0 small enough, this problem has a positive solution vε and that these solutions concentrate and blow up along the sphere as ε tends to 0. Our approach is to reduce this problem to a critical problem of the form -Δu=Q(x)|u|4/(n-2)u in Ωε, u=0 on ∂Ωε, in a punctured domain Ωε:=\x∈Ω:|x-ξ0|>ε\ of lower dimension, by means of some Hopf map. We show that, if Ω is a bounded smooth domain in ℝn, n≥3, ξ0 is inΩ, Q is in C2(\b\Oarmega) is positive and ∇ Q(ξ0)≠0 then, for ε>0 small enough, this problem has a positive solution uε, and that these solutions concentrate and blow up at ξ0 as ε goes to 0.