A note on a sinh-Poisson type equation with variable intensities on pierced domains

We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain \begin{equation*} \left\{ \begin{array}{ll} \Delta u +\rho\left(V_1(x)e^{u}- V_2(x)e^{-\tau u}\right)=0 &\text{in } \Omega_\epsilon:=\Omega\setminus \displaystyle \bigcup_{i=1}^m \overline{B(\xi_i,\epsilon_i)}\\ u=0&\text{on }\partial\Omega_\epsilon, \end{array}\right. \end{equation*} where $\rho>0$, $V_1,V_2>0$ are smooth potentials in $\Omega$, $\tau>0$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$ and $B(\xi_i,\epsilon_i)$ is a ball centered at $\xi_i\in \Omega$ with radius $\epsilon_i>0$, $i=1,\dots,m$. When $\rho>0$ is small enough and $m_1\in \{1,\dots,m-1\}$, there exist radii $\epsilon=(\epsilon_1,\dots,\epsilon_m)$ small enough such that the problem has a solution which blows-up positively at the points $\xi_1,\dots,\xi_{m_1}$ and negatively at the points $\xi_{m_1+1},\dots,\xi_{m}$ as $\rho\to 0$. The result remains true in cases $m_1=0$ with $V_1\equiv 0$ and $m_1=m$ with $V_2\equiv 0$, which are Liouville type equations.