  We study the mean-field limits of critical points of interaction energies
with Coulombian singularity. An important feature of our setting is that we
allow interaction between particles of opposite signs. Particles of opposite
signs attract each other whereas particles of the same signs repel each other.
In 2D, we prove that the associated empirical measures converge to a limiting
measure $\mu$ that satisfies a two-fold criticality condition: in velocity form
or in vorticity form. Our setting includes the stationary attraction-repulsion
problem with Coulombian singularity and the stationary system of point-vortices
in fluid mechanics. In this last context, in the case where the limiting
measure is in $H^{-1}_{\text{loc}}({\mathbb R}^2)$, we recover the classical
criticality condition stating that $\nabla^\perp g \ast \mu$, with $g(x)=-\log
|x|$, is a stationary solution of the incompressible Euler equation. This
result, is, to the best of our knowledge, new in the case of particles with
different signs (for particles of the positive sign it was obtained by Schochet
in 1996). In order to derive the limiting criticality condition in the velocity
form, we follow an approach devised by Sandier-Serfaty in the context of
Ginzburg-Landau vortices. This consists of passing to the limit in the
stress-energy tensor associated with the velocity field. On the other hand, the
criticality condition in the vorticity form is obtained by arguments closer to
the ones of Schochet.
