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We consider a class of variational equations with exponential nonlinearitieson a compact Riemannian surface, describing the mean field equation of theequilibrium turbulance with arbitrarily signed vortices. For the first time, weconsider the problem with both supercritical parameters and we give anexistence result by using variational methods. In doing this, we present a newMoser-Trudinger type inequality under suitable conditions on the center of massand the scale of concentration of both e^u and e^{-u}, where u is the unknownfunction in the equation.

An Existence Result for the Mean Field Equation on Compact Surfaces in a Doubly Supercritical Regime