Dynamics of threshold solutions for the energy-critical inhomogeneous NLS

TL;DR

This paper analyzes the long-term behavior of solutions at the energy threshold for the focusing inhomogeneous energy-critical Schrödinger equation, classifying their dynamics in dimensions 3 to 5.

Contribution

It constructs special threshold solutions and classifies all solutions at the energy threshold, extending understanding of their long-time dynamics in inhomogeneous settings.

Findings

Existence of special threshold solutions $W^\\pm$ approaching the ground state $W$.

Solutions at threshold energy either behave like subthreshold solutions or match $W, W^+$, or $W^-$.

Classification of threshold solutions in dimensions 3, 4, and 5.

Abstract

In this article, we study the long-time dynamics of threshold solutions for the focusing energy-critical inhomogeneous Schr\"odinger equation and classify the corresponding threshold solutions in dimensions $d=3,4,5$. We first show the existence of special threshold solutions $W^\pm$ by constructing a sequence of approximate solutions in suitable Lorentz space, which exponentially approach the ground state $W$ in one of the time directions. We then prove that solutions with threshold energy either behave as in the subthreshold case or agree with $W, W^+$ or $W^-$ up to the symmetries of the equation. The proof relies on detailed spectral analysis of the linearized Schr\"odinger operator, the relevant modulation analysis, the global Virial analysis, and the concentration compactness argument in the Lorentz space.