Inserted: 15 sep 2016
Last Updated: 6 mar 2017
Journal: J. Diff. Eq.
A class of equations with exponential nonlinearities on a compact Riemannian surface is considered. More precisely, we study an asymmetric sinh-Gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with variable intensities.
We start by performing a blow-up analysis in order to derive some information on the local blow-up masses. As a consequence we get a compactness property in a supercritical range.
We next introduce a variational argument based on improved Moser-Trudinger inequalities which yields existence of solutions for any choice of the underlying surface.
Keywords: Variational methods, blow-up analysis, Geometric PDEs, Mean field equation