Mass quantization and minimax solutions for Neri's mean field equation in 2D-turbulence

We study the mean field equation derived by Neri in the context of the statistical mechanics description of 2D-turbulence, under a ``stochastic" assumption on the vortex circulations. The corresponding mathematical problem is a nonlocal semilinear elliptic equation with exponential type nonlinearity, containing a probability measure $mathcal Pinmathcal M([-1,1])$ which describes the distribution of the vortex circulations. Unlike the more investigated ``deterministic" version, we prove that Neri's equation may be viewed as a perturbation of the widely analyzed standard mean field equation, obtained by taking $mathcal P=delta_1$. In particular, in the physically relevant case where $mathcal P$ is non-negatively supported and $mathcal P({1})>0$, we prove the mass quantization for blow-up sequences. We apply this result to construct minimax type solutions on bounded domains in $mathbb R^2$ and on compact 2-manifolds without boundary.