*Accepted Paper*

**Inserted:** 11 feb 2016

**Last Updated:** 8 sep 2016

**Journal:** Indiana Univ. Math. J.

**Year:** 2016

**Abstract:**

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 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 $\rho_2 \notin 8\pi\mathbb{N}$.

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