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

Blow-up for the pointwise NLS in dimension two: absence of critical power