  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.
