New existence results for the mean field equation on compact surfaces via degree theory

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.