*Accepted Paper*

**Inserted:** 25 mar 2015

**Last Updated:** 25 mar 2015

**Journal:** Rend. Semin. Mat. Univ. Padova

**Year:** 2014

**Abstract:**

We consider the following class of equations with exponential nonlinearities on a closed surface which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices.

By considering the parity of the Leray-Schauder degree associated to the problem, we prove solvability for $\rho_i \in (8\pi k, 8\pi(k+1)),\, k \in \mathbb{N}$. Our theorem provides a new existence result in the case when the underlying manifold is a sphere and gives a completely new proof for other known results.

**Download:**