Extremal boundedness of a variational functional in point vortex mean   field theory associated with probability measures
  We study a variational functional of Trudinger-Moser type associated with one-sided Borel probability measure. Its boundedness at the extremal parameter holds when the residual vanishing occurs. In the proof we use a variant of the Y.Y. Li estimate. 
Takashi Suzuki, Ryo Takahashi and Xiao Zhang
math.AP