  We consider the Schr\"odinger equation in dimension two with a fixed,
pointwise, focusing nonlinearity and show the occurrence of a blow-up
phenomenon with two peculiar features: first, the energy threshold under which
all solutions blow up is strictly negative and coincides with the infimum of
the energy of the standing waves. Second, there is no critical power
nonlinearity, i.e. for every power there exist blow-up solutions. This last
property is uncommon among the conservative Schr\"odinger equations with local
nonlinearity.
