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 $$left{ begin{array}{ll} -Delta u=lambda_1dfrac{V_1 e^{u}}{ int_{Omega_{boldsymbolepsilon}} V_1 e^{u} dx } - lambda_2tau dfrac{ V_2 e^{-tau u}}{ int_{Omega_{boldsymbolepsilon}}V_2 e^{ - tau u} dx}&text{in $Omega_{boldsymbolepsilon}=Omegasetminus displaystyle bigcup_{i=1}^m overline{B(xi_i,epsilon_i)}$} u=0 &text{on $partial Omega_{boldsymbolepsilon}$}, end{array} right. $$ where $B(xi_i,epsilon_i)$ is a ball centered at $xi_iinOmega$ with radius $epsilon_i$, $tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $lambda_1>8pi m_1$ and $lambda_2 tau^2>8pi (m-m_1)$ with $m_1 in {0,1,dots,m}$, there exist radii $epsilon_1,dots,epsilon_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $xi_1,dots,xi_{m_1}$ and $xi_{m_1+1},dots,xi_{m}$, respectively, as the radii approach zero.