Global existence and blowup for a class of the focusing nonlinear Schr\"odinger equation with inverse-square potential

TL;DR

This paper investigates the global existence and blowup phenomena for a class of focusing nonlinear Schrödinger equations with inverse-square potential across various critical regimes, extending previous results to higher dimensions and full parameter ranges.

Contribution

It extends known results on global behavior and blowup for focusing NLS with inverse-square potential to all dimensions d≥3 and a full range of the parameter c, including critical cases.

Findings

Proves global existence and blowup below ground states in the mass-critical case for d≥3.

Establishes global existence and blowup below ground state threshold in intercritical case.

Demonstrates blowup below ground states in the energy-critical case for certain c values.

Abstract

We consider a class of the focusing nonlinear Schr\"odinger equation with inverse-square potential \[ i\partial_t u + \Delta u -c|x|^{-2}u = - |u|^\alpha u, \quad u(0)=u_0 \in H^1, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $\frac{4}{d}\leq \alpha \leq \frac{4}{d-2}$ and $c\ne 0$ satisfies $c>-\lambda(d):=-\left(\frac{d-2}{2}\right)^2$. In the mass-critical case $\alpha=\frac{4}{d}$, we prove the global existence and blowup below ground states for the equation with $d\geq 3$ and $c>-\lambda(d)$. In the mass and energy intercritical case $\frac{4}{d}<\alpha<\frac{4}{d-2}$, we prove the global existence and blowup below the ground state threshold for the equation. This extends similar results of \cite{KillipMurphyVisanZheng} and \cite{LuMiaoMurphy} to any dimensions $d\geq 3$ and a full range $c>-\lambda(d)$. We finally prove the blowup below ground states for the equation in the energy-critical case $\alpha=\frac{4}{d-2}$ with $d\geq 3$ and $c>-\frac{d^2+4d}{(d+2)^2} \lambda(d)$.