On the Topological degree of the Mean field equation with two parameters

We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $$ - \Delta u = \rho_1 \left( \frac{h_1 \,e^{u}}{\int_M h_1 \,e^{u} } - \frac{1}{|M|} \right) - \rho_2 \left( \frac{h_2 \,e^{-u}}{\int_M h_2 \,e^{-u} } - \frac{1}{|M|} \right), $$ which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h_1, h_2$ are smooth positive functions and $\rho_1, \rho_2$ are two positive parameters. We start by proving a concentration phenomena for the above equation, which leads to a-priori bound for the solutions of this problem provided $\rho_i\notin 8\pi\mathbb{N}, \, i=1,2$. Then we study the blow up behavior when $\rho_1$ crosses $8\pi$ and $\rho_2 \notin 8\pi\mathbb{N}$. By performing a suitable decomposition of the above equation and using the shadow system that was introduced for the $SU(3)$ Toda system, we can compute the Leray-Schauder topological degree for $\rho_1 \in (0,8\pi) \cup (8\pi,16\pi)$ and $\rho_2 \notin 8\pi\mathbb{N}$. As a byproduct our argument, we give new existence results when the underlying manifold is a sphere and a new proof for some known existence result.