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A class of equations with exponential nonlinearities on a compact Riemanniansurface is considered. More precisely, we study an asymmetric sinh-Gordonproblem arising as a mean field equation of the equilibrium turbulence ofvortices with variable intensities.  We start by performing a blow-up analysis in order to derive some informationon the local blow-up masses. As a consequence we get a compactness property ina supercritical range.  We next introduce a variational argument based on improved Moser-Trudingerinequalities which yields existence of solutions for any choice of theunderlying surface.

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