# 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 $Σ$:

$−Δu=ρ_{1}(∫_{Σ}he_{u}dV_{g}he_{u} −∣Σ∣1 )−ρ_{2}(∫_{Σ}he_{−u}dV_{g}he_{−u} −∣Σ∣1 ),$

which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h$ is a smooth positive function and $ρ_{1},ρ_{2}$ two positive parameters. By considering the parity of the Leray–Schauder degree associated to the problem, we prove solvability for $ρ_{i}∈(8πk,8π(k+1)),k∈N$. 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