  This paper examines the decay properties of positive solutions for a family
of fully nonlinear systems of integral equations containing Wolf potentials and
Hardy weights. This class of systems includes examples which are closely
related to the Euler-Lagrange equations for several classical inequalities such
as the Hardy-Sobolev and Hardy-Littlewood-Sobolev inequalities. In particular,
a complete characterization of the fast decaying ground states in terms of
their integrability is provided in that bounded and fast decaying solutions are
shown to be equivalent to the integrable solutions. In generating this
characterization, additional properties for the integrable solutions, such as
their boundedness and optimal integrability, are also established. Furthermore,
analogous decay properties for systems of quasilinear equations of the weighted
Lane-Emden type are also obtained.
