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We are concerned with an elliptic problem which describes a mean fieldequation of the equilibrium turbulence of vortices with variable intensities.  In the first part of the paper we describe the blow-up phenomenon andhighlight the differences from the standard mean field equation.  In the second part we discuss the Moser-Trudinger inequality in terms of theblow-up masses and get the existence of solutions in a non-coercive regime bymeans of a variational argument, which is based on some improvedMoser-Trudinger inequalities.

A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects