  In this paper we will continue the analysis of two dimensional Schr\"odinger
equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this
model, the occurrence of a blow-up phenomenon has two peculiar features: the
energy threshold under which all solutions blow up is strictly negative and
coincides with the infimum of the energy of the standing waves; there is no
critical power nonlinearity, i.e., for every power there exist blow-up
solutions. Here we study the stability properties of stationary states to
verify whether the anomalies mentioned before have any counterpart on the
stability features.
