  We prove sharp density upper bounds on optimal length-scales for the ground
states of classical 2D Coulomb systems and generalizations thereof. Our method
is new, based on an auxiliary Thomas-Fermi-like variational model. Moreover, we
deduce density upper bounds for the related low-temperature Gibbs states. Our
motivation comes from fractional quantum Hall physics, more precisely, the
perturbation of the Laughlin state by external potentials or impurities. These
give rise to a class of many-body wave-functions that have the form of a
product of the Laughlin state and an analytic function of many variables. This
class is related via Laughlin's plasma analogy to Gibbs states of the
generalized classical Coulomb systems we consider. Our main result shows that
the perturbation of the Laughlin state cannot increase the particle density
anywhere, with implications for the response of FQHE systems to external
perturbations.
