Calculus of Variations and Geometric Measure Theory

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

Stability of the standing waves of the concentrated NLSE in dimension two

created by tentarelli on 07 Jul 2022


Published Paper

Inserted: 7 jul 2022
Last Updated: 7 jul 2022

Journal: Mathematics in Engineering
Volume: 3
Number: 2
Pages: 1-15
Year: 2021
Doi: 10.3934/mine.2021011

ArXiv: 2001.03969 PDF

Special Issue “Nonlinear models in applied mathematics”.


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.