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 $$\{\begin{array}{ll}-\Delta u={\lambda }_1\frac{V_1{e}^u}{{\int }_{{\Omega }_{\mathbf{\epsilon }}}{V}_1{e}^{u}dx}-{\lambda }_2\tau \frac{V_2{e}^{-\tau u}}{{\int }_{{\Omega }_{\mathbf{\epsilon }}}{V}_2{e}^{-\tau u}dx}& \text{in Ωε=Ω∖i=1⋃mB(ξi,εi)}\\ u=0& \text{on ∂Ωε},\end{array}$$ 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.

23 pages