Calculus of Variations and Geometric Measure Theory

R. Adami - R. Carlone - M. Correggi - L. Tentarelli

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

created by tentarelli on 07 Jul 2022


Published Paper

Inserted: 7 jul 2022
Last Updated: 7 jul 2022

Journal: J. Differential Equations
Volume: 269
Number: 1
Pages: 1-37
Year: 2020
Doi: 10.1016/j.jde.2019.11.096

ArXiv: 1808.10343 PDF


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.