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

Abstract

We consider the Schrödinger 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 Schrödinger equations with local nonlinearity.

arXiv comment29 pages, in press on J. Differential Equations (some changes in the structure of the paper have been made with respect to the previous version). Keywords: NLS, blow up, point interactions, standing waves

References in corpus (15)

• On Schroedinger Equations with Concentrated Nonlinearities
• Small data blow-up of L^{2}-solution for the nonlinear Schrödinger equation without gauge invariance
• The point-like limit for a NLS equation with concentrated nonlinearity in dimension three
• The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit
• The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions
• The fractional Schrödinger equation with singular potential and measure data
• Coherent patterns and self-induced diffraction of electrons on a thin nonlinear layer
• Well-posedness of the Two-dimensional Nonlinear Schrödinger Equation with Concentrated Nonlinearity
• On the nonexistence of Green's function and failure of the strong maximum principle
• Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory
• Two-dimensional Time-dependent Point Interactions
• Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one
• Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles
• The 1-D Dirac equation with concentrated nonlinearity
• Potential capacity and applications

Cited by in corpus (15)

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• Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect
• Blow-up and instability of standing waves for the NLS with a point interaction in dimension two
• Stability of the standing waves of the concentrated NLSE in dimension two
• Complete ionization for a non-autonomous point interaction model in d = 2
• A general review on the NLS equation with point-concentrated nonlinearity
• Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
• Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
• Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions
• On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction
• Enhancement of the Zakharov-Glassey's method for Blow-Up in nonlinear Schroedinger equations
• Scattering for the $L^2$ supercritical point NLS
• Non-Kirchhoff Vertices and NLS Ground States on graphs