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 $$\left\{ \begin{array}{ll} -Δu=λ_1\dfrac{V_1 e^{u}}{ \int_{Ω_{\boldsymbolε}} V_1 e^{u} dx } - λ_2τ\dfrac{ V_2 e^{-τu}}{ \int_{Ω_{\boldsymbolε}}V_2 e^{ - τu} dx}&\text{in $Ω_{\boldsymbolε}=Ω\setminus \displaystyle \bigcup_{i=1}^m \overline{B(ξ_i,ε_i)}$}\\ \ \ u=0 &\text{on $\partial Ω_{\boldsymbolε}$}, \end{array} \right. $$ where $B(ξ_i,ε_i)$ is a ball centered at $ξ_i\inΩ$ with radius $ε_i$, $τ$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $λ_1>8πm_1$ and $λ_2 τ^2>8π(m-m_1)$ 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 $ξ_1,\dots,ξ_{m_1}$ and $ξ_{m_1+1},\dots,ξ_{m}$, respectively, as the radii approach zero.