  The equations of electrostatics are presented in pre-metric form, and it is
pointed out that if the origin of the nonlinearity is the constitutive law for
the medium then the differential equations themselves remain linear, while the
nonlinearity is confined to an algebraic equation. These equations are solved
for a general class of electric fields that include the common textbook
examples, namely, fields that are adapted to a coordinate vector field. The
special forms that they then take for particular electric constitutive laws of
quantum origin, namely, the constitutive laws derived from the Born-Infeld and
Heisenberg-Euler Lagrangians, are then discussed. Finally, the classical
problem of modeling the electron is redefined in light of the established facts
of quantum physics.
