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 $$\{\begin{array}{ll}-\mathrm{\Delta }u={\lambda }_1{\displaystyle \frac{V_1{e}^u}{{\int }_{{\mathrm{\Omega }}_{\mathit{ϵ}}}{V}_1{e}^{u}dx}}-{\lambda }_2\tau {\displaystyle \frac{V_2{e}^{-\tau u}}{{\int }_{{\mathrm{\Omega }}_{\mathit{ϵ}}}{V}_2{e}^{-\tau u}dx}}& \text{in }{\mathrm{\Omega }}_{\mathit{ϵ}}=\mathrm{\Omega }\setminus {\displaystyle \bigcup _{i=1}^m\overline{B({\xi }_i,{ϵ}_i)}}\\ u=0& \text{on }\mathrm{\partial }{\mathrm{\Omega }}_{\mathit{ϵ}},\end{array}$$ where $B({\xi }_i,{ϵ}_i)$ is a ball centered at ${\xi }_i\in \mathrm{\Omega }$ with radius ${ϵ}_i$, $\tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When ${\lambda }_1>8\pi m_1$ and ${\lambda }_2{\tau }^2>8\pi (m-m_1)$ with $m_1\in \{0,1,\dots ,m\}$, there exist radii ${ϵ}_1,\dots ,{ϵ}_m$ 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.