  We consider a class of variational equations with exponential nonlinearities
on a compact Riemannian surface, describing the mean field equation of the
equilibrium turbulance with arbitrarily signed vortices. For the first time, we
consider the problem with both supercritical parameters and we give an
existence result by using variational methods. In doing this, we present a new
Moser-Trudinger type inequality under suitable conditions on the center of mass
and the scale of concentration of both e^u and e^{-u}, where u is the unknown
function in the equation.
