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In this paper we consider the minimizing sequence for some energy functionalof an elliptic equation associated with the mean field limit of the pointvortex distribution one-sided Borel probability measure. If such a sequenceblows up, we derive some estimate which is related to the behavior of solutionnear the blow-up point. Moreover, we study the two intensities case to considerthe sufficient condition for this estimate. Our main results are new for thestandard mean field equation as well.

Blow up analysis for Boltzmann-Poisson equation in Onsager's theory for point vortices with multi-intensities