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

Abstract

We consider the following class of equations with exponential nonlinearities on a closed surface $\Sigma$:

which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h$ is a smooth positive function and ${\rho }_1,{\rho }_2$ two positive parameters. 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.