On the mean field equation with variable intensities on pierced domains. (January 2020)

• Record Type: Journal Article
• Authors: Esposito, Pierpaolo; Figueroa, Pablo; Pistoia, Angela
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: 35B44 -- 35J25 -- 35J60; Pierced domain -- Blowing-up solutions -- Mean field equation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111597
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain − Δ u = λ 1 V 1 e u ∫ Ω ϵ V 1 e u d x − λ 2 τ V 2 e − τ u ∫ Ω ϵ V 2 e − τ u d x in Ω ϵ = Ω ∖ ⋃ i = 1 m B ( ξ i, ϵ i ) ¯ u = 0 on ∂ Ω ϵ, where B ( ξ i, ϵ i ) is a ball centered at ξ i ∈ Ω with radius ϵ i, τ is a positive parameter and V 1, V 2 > 0 are smooth potentials. When λ 1 > 8 π m 1 and λ 2 τ 2 > 8 π ( m − m 1 ) with m 1 ∈ { 0, 1, …, m }, there exist radii ϵ 1, …, ϵ m small enough such that the problem has a solution which blows-up positively and negatively at the points ξ 1, …, ξ m 1 and ξ m 1 + 1, …, ξ m, respectively, as the radii approach zero.