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. Here is a smooth positive function and two positive parameters. By considering the parity of the Leray–Schauder degree associated to the problem, we prove solvability for . 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. 11–17DOI 10.4171/RSMUP/136-2