Minimal mass blow-up solutions for the $L^2$ critical NLS with inverse-square potential

Abstract

We study minimal mass blow-up solutions of the focusing $L^2$ critical nonlinear Schrödinger equation with inverse-square potential, \[ i\partial_t u + Δu + \frac{c}{|x|^2}u+|u|^{\frac{4}{N}}u = 0, \] with $N⩾3$ and $0<c<\frac{(N-2{)}^2}{4}$. We first prove a sharp global well-posedness result: all $H^1$ solutions with a mass (i.e. $L^2$ norm) strictly below that of the ground states are global. Note that, unlike the equation in free space, we do not know if the ground state is unique in the presence of the inverse-square potential. Nevertheless, all ground states have the same, minimal, mass. We then construct and classify finite time blow-up solutions at the minimal mass threshold. Up to the symmetries of the equation, every such solution is a pseudo-conformal transformation of a ground state solution.

arXiv commentJournal version, references added, minor changes including the definition of ground states, Propositions 1 and 3, Remarks 1 and 2

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Cited by in corpus (6)

• On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential
• Minimal mass blow-up solutions for nonlinear Schrödinger equations with an inverse potential
• Remarks on minimal mass blow up solutions for a double power nonlinear Schrödinger equation
• Scattering and blow-up criteria for 3D cubic focusing nonlinear inhomogeneous NLS with a potential
• Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potential
• On the blow up phenomenon for the mass critical focusing Hartree equation with inverse-square potential