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. When λ1>8πm1 and λ2τ2>8π(m−m1) with m1∈{0,1,…,m}, there exist radii ϵ1,…,ϵm small enough such that the problem has a solution which blows-up positively and negatively at the points ξ1,…,ξmjavax.xml.bind.JAXBElement@b919342 and ξmjavax.xml.bind.JAXBElement@186c9427+1,…,ξm, respectively, as the radii approach zero.