Threshold solutions for the $3d$ cubic INLS: the energy-critical case

TL;DR

This paper investigates special threshold solutions for the energy-critical 3D cubic inhomogeneous nonlinear Schrödinger equation, characterizing solution behaviors at the ground state energy level with novel analytical techniques.

Contribution

It establishes the existence of special solutions at the ground state energy and introduces a new modulation analysis approach due to the singular nonlinearity.

Findings

Existence of solutions $W^ ext{±}$ at ground state energy

Characterization of solution dynamics at critical energy level

Development of a novel modulation analysis method

Abstract

We study the energy-critical $3d$ cubic inhomogeneous NLS equation $i\partial_t u + \Delta u + |x|^{-1}|u|^2 u=0$. In this work, we prove the existence of special solutions $W^\pm$ with energy equal to that of the ground state $W$ and use these solutions to characterize the behavior of solutions at the ground state energy. The singular factor $|x|^{-1}$ in the nonlinearity significantly limits the smoothness of the ground state and prompts a novel approach to the modulation analysis.