On the mean field equation with variable intensities on pierced domains

Abstract

We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$\left\{ \begin{array}{ll} -\Delta u=\lambda_1\dfrac{V_1 e^{u}}{ \int_{\Omega_{\boldsymbol\epsilon}} V_1 e^{u} dx } - \lambda_2\tau \dfrac{ V_2 e^{-\tau u}}{ \int_{\Omega_{\boldsymbol\epsilon}}V_2 e^{ - \tau u} dx}&\text{in $\Omega_{\boldsymbol\epsilon}=\Omega\setminus \displaystyle \bigcup_{i=1}^m \overline{B(\xi_i,\epsilon_i)}$}\\ \ \ u=0 &\text{on $\partial \Omega_{\boldsymbol\epsilon}$}, \end{array} \right. $$ where $B(\xi_i,\epsilon_i)$ is a ball centered at $\xi_i\in\Omega$ with radius $\epsilon_i$, $\tau$ 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 $\epsilon_1,\dots,\epsilon_m$ 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.