Calculus of Variations and Geometric Measure Theory

A. Jevnikar

Blow-up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities

created by jevnikar on 15 Sep 2016
modified on 06 Mar 2017


Accepted Paper

Inserted: 15 sep 2016
Last Updated: 6 mar 2017

Journal: J. Diff. Eq.
Year: 2017


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