  We study how a neutralising cloud of counterions screens the electric field
of a uniformly charged planar membrane plate, when the counterions are
characterised by a distribution of charges (or valence), $n(q)$. We work out
analytically the one-plate and two-plate cases, at the level of non-linear
Poisson-Boltzmann theory. The (essentially asymptotic) predictions are
successfully compared to numerical solutions of the full Poisson-Boltzmann
theory, but also to Monte Carlo simulations. The counterions with smallest
valence control the long-distance features of interactions, and may
qualitatively change the results pertaining to the classic monodisperse case
where all counterions have the same charge. Emphasis is put on continuous
distributions $n(q)$, for which new power-laws can be evidenced, be it for the
ionic density or the pressure, in the one- and two-plates situations
respectively. We show that for discrete distributions, more relevant for
experiments, these scaling laws persist in an intermediate but yet observable
range. Furthermore, it appears that from a practical point of view, hallmarks
of the continuous $n(q)$ behaviour is already featured by discrete mixtures
with a relatively small number of constituents.
