Liouville Systems of Mean Field Equations

In this article, we discuss the recent work of Lin and Zhang on the Liouville system of mean field equations: $$Delta{u}_i+sum_{j}a_{ij}rho_{j} ({frac{{h_j}e^{u_{j}}}{int_{M}{h_{j}e^{u_{j}}}}-{frac{1}{|M|}}})=0,, quad{rm on}, M,$$ where M is a compact Riemann surface and |M| is the area, or $$Delta{u}_i+sum_{j}a_{ij}rho_{j} frac{{h_j}e^{u_{j}}}{int_{Omega}{h_{j}e^{u_{j}}}}=0,, quad{rm in}, Omega,$$ $${u_j}=0,,, quad{rm on}, partialOmega, $$ where ?? is a bounded domain in ${mathbb{R}^2}$ . Among other things, we completely determine the set of non-critical parameters and derive a degree counting formula for these systems.