On the mean field equation with variable intensities on pierced domains

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.