  We consider a class of equations with exponential non-linearities on a
compact surface which arises as the mean field equation of the equilibrium
turbulence with arbitrarily signed vortices. We prove an existence result via
degree theory. This yields new existence results in case of a topological
sphere. The proof is carried out by considering the parity of the
Leray-Schauder degree associated to the problem. With this method we recover
also some known previous results.
