Dynamics of threshold solutions for energy critical NLS with inverse square potential

TL;DR

This paper analyzes the behavior of solutions to the energy critical nonlinear Schrödinger equation with inverse square potential, classifying scattering, blow-up, and convergence to ground states across different dimensions.

Contribution

It provides a detailed characterization of solution dynamics near the ground state for the energy critical NLS with inverse square potential, including stability and blow-up results.

Findings

Solutions with less kinetic energy than the ground state scatter or converge to it.

Radial solutions with greater kinetic energy blow up in finite time.

Exponential convergence to the ground state in certain cases.

Abstract

We consider the focusing energy critical NLS with inverse square potential in dimension $d= 3, 4, 5$ with the details given in $d=3$ and remarks on results in other dimensions. Solutions on the energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifolds of the ground state. In the latter case they converge to the ground state exponentially in the energy space as $t\to \infty$ or $t\to -\infty$. (In 3-dim without radial assumption, this holds under the compactness assumption of non-scattering solutions on the energy surface.) When the kinetic energy is greater than that of the ground state, we show that all radial $H^1$ solutions blow up in finite time, with the only two exceptions in the case of 5-dim which belong to the stable/unstable manifold of the ground state. The proof relies on the detailed spectral analysis, local invariant manifold theory, and a global Virial analysis.