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

TL;DR

This paper investigates an asymmetric mean field equation with exponential nonlinearities on compact surfaces, analyzing blow-up behavior and establishing solution existence in the supercritical regime using variational methods.

Contribution

It provides a detailed blow-up analysis and proves the existence of solutions in the supercritical case for an asymmetric sinh-Gordon problem with variable vortex intensities.

Findings

Derived local blow-up masses for the equation

Established a compactness property in the supercritical range

Proved existence of solutions on any compact surface

Abstract

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.