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

### Aleks Jevnikar

Università di Roma 'Tor Vergata', Italy

## 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.

## Cite this article

Aleks Jevnikar, New existence results for the mean field equation on compact surfaces via degree theory. Rend. Sem. Mat. Univ. Padova 136 (2016), pp. 11–17

DOI 10.4171/RSMUP/136-2