  We solve a nonlinear integral equation for the electrostatic potential in
doped graphene due to an external charge, arising from a Thomas-Fermi (TF)
model for screening by graphene's $\pi$ electron bands. In particular, we study
the effects of a finite equilibrium charge carrier density in graphene,
non-zero temperature, non-zero gap between graphene and a dielectric substrate,
as well as the nonlinearity in the band density of states. Effects of the
exchange and correlation interactions are also briefly discussed for undoped
graphene at zero temperature. Nonlinear results are compared with both the
linearized TF model and the dielectric screening model within random phase
approximation (RPA). In addition, image potential of the external charge is
evaluated from the solution of the nonlinear integral equation and compared to
the results of linear models. We have found generally good agreement between
the results of the nonlinear TF model and the RPA model in doped graphene,
apart from Friedel oscillations in the latter model. However, relatively strong
nonlinear effects are found in the TF model to persist even at high doping
densities and large distances of the external charge.
