  An expression for the density of a planar induced vacuum charge is obtained
in a strong Coulomb potential in coordinate space. Treatment is based on a
self-adjoint extension approach for constructing of the Green's function of a
charged fermion in this potential. Induced vacuum charge density is calculated
and analyzed at the subcritical and supercritical Coulomb potentials for
massless and massive fermions. The behavior of the obtained vacuum charge
density is investigated at long and short distances from the Coulomb center.
The induced vacuum charge has a screening sign. Screening of a Coulomb impurity
in graphene is briefly discussed. We calculate the real vacuum polarization
charge density that acquires the quantum electrodynamics vacuum in the
supercritical Coulomb potential due to the so-called real vacuum polarization.
It is shown that the vacuum charge densities essentially differ in massive and
massless cases. We expect that our results can, as a matter of principle, be
tested in graphene with a supercritical Coulomb impurity.
