Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence

We consider the mean field equation arising in the high-energy scaling limit of point vortices with a general circulation constraint, when the circulation number density is subject to a probability measure. Mathematically, such an equation is a non-local elliptic equation containing an exponential nonlinearity which depends on this probability measure. We analyze the behavior of blow-up sequences of solutions in relation to the circulation numbers. As an application of our analysis we derive an improved Trudinger-Moser inequality for the associated variational functional.