Calculus of Variations and Geometric Measure Theory

A. Jevnikar

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

created by jevnikar on 25 Mar 2015


Accepted Paper

Inserted: 25 mar 2015
Last Updated: 25 mar 2015

Journal: Rend. Semin. Mat. Univ. Padova
Year: 2014


We consider the following class of equations with exponential nonlinearities on a closed surface which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices.

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.