On the mean field equation with variable intensities on pierced domains
We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain −Δu=λ1 [Formula presented] −λ2τ [Formula presented] in Ωϵ=Ω∖⋃i=1mB(ξi,ϵi)¯u=0on ∂Ωϵ, where B(ξi,ϵi) is a ball centered at ξi∈Ω with radius ϵi, τ is a positive parameter and V1,V2>0 are smooth potentials.