  We propose a density functional theory of Thomas-Fermi-Dirac-von Weizs\"acker
type to describe the response of a single layer of graphene resting on a
dielectric substrate to a point charge or a collection of charges some distance
away from the layer. We formulate a variational setting in which the proposed
energy functional admits minimizers, both in the case of free graphene layers
and under back-gating. We further provide conditions under which those
minimizers are unique and correspond to configurations consisting of
inhomogeneous density profiles of charge carrier of only one type. The
associated Euler-Lagrange equation for the charge density is also obtained, and
uniqueness, regularity and decay of the minimizers are proved under general
conditions. In addition, a bifurcation from zero to non-zero response at a
finite threshold value of the external charge is proved.
