On the mean field equation with variable intensities on pierced domains. (January 2020) - Esposito, Pierpaolo; Figueroa, Pablo; Pistoia, Angela
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.