  We complement a recent exact study by L. Samaj on the properties of a guest
charge $Q$ immersed in a two-dimensional electrolyte with charges $+1/-1$. In
particular, we are interested in the behavior of the density profiles and
electric potential created by the charge and the electrolyte, and in the
determination of the renormalized charge which is obtained from the
long-distance asymptotics of the electric potential. In Samaj's previous work,
exact results for arbitrary coulombic coupling $\beta$ were obtained for a
system where all the charges are points, provided $\beta Q<2$ and $\beta < 2$.
Here, we first focus on the mean field situation which we believe describes
correctly the limit $\beta\to 0$ but $\beta Q$ large. In this limit we can
study the case when the guest charge is a hard disk and its charge is above the
collapse value $\beta Q>2$. We compare our results for the renormalized charge
with the exact predictions and we test on a solid ground some conjectures of
the previous study. Our study shows that the exact formulas obtained by Samaj
for the renormalized charge are not valid for $\beta Q>2$, contrary to a
hypothesis put forward by Samaj. We also determine the short-distance
asymptotics of the density profiles of the coions and counterions near the
guest charge, for arbitrary coulombic coupling. We show that the coion density
profile exhibit a change of behavior if the guest charge becomes large enough
($\beta Q\geq 2-\beta$). This is interpreted as a first step of the counterion
condensation (for large coulombic coupling), the second step taking place at
the usual Manning--Oosawa threshold $\beta Q=2$.
