  We study the Cauchy problem for nonlinear Schr\"odinger equations with
attractive inverse-power potentials. By using variational arguments, we first
determine a sharp threshold of global well-posedness and blow-up for the
equation in the mass-supercritical case. We next study the existence and
non-existence of minimizers for the energy functional with prescribed mass
constraint. In the mass-critical case, we also study the blow-up behavior of
minimizers when the mass tends to a critical value.
