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

Luccas Campos, Luiz Gustavo Farah, Jason Murphy

TL;DR

This work addresses threshold dynamics for the energy-critical 3D INLS with singular nonlinearity $|x|^{-1}|u|^2u$. It develops a refined modulation framework that uses a low-frequency projection of the ground state to overcome lack of smoothness, together with Lin–Zeng spectral analysis to identify a single unstable direction. The authors prove the existence of two threshold solutions $W^{\pm}$ at energy $E[W]$ and establish exponential convergence to the ground state, enabling a complete classification of threshold dynamics (scattering, convergence to $W$, and the conjectured blow-up scenario for $W^+$). These results extend the threshold dynamics theory to energy-critical INLS in 3D and introduce a robust modulation strategy for highly singular nonlinearities, with potential applicability to other singular dispersive equations.

Abstract

We study the energy-critical $3d$ cubic inhomogeneous NLS equation $i\partial_t u + Δu + |x|^{-1}|u|^2 u=0$. 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.

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

TL;DR

This work addresses threshold dynamics for the energy-critical 3D INLS with singular nonlinearity . It develops a refined modulation framework that uses a low-frequency projection of the ground state to overcome lack of smoothness, together with Lin–Zeng spectral analysis to identify a single unstable direction. The authors prove the existence of two threshold solutions at energy and establish exponential convergence to the ground state, enabling a complete classification of threshold dynamics (scattering, convergence to , and the conjectured blow-up scenario for ). These results extend the threshold dynamics theory to energy-critical INLS in 3D and introduce a robust modulation strategy for highly singular nonlinearities, with potential applicability to other singular dispersive equations.

Abstract

We study the energy-critical cubic inhomogeneous NLS equation . In this work, we prove the existence of special solutions with energy equal to that of the ground state and use these solutions to characterize the behavior of solutions at the ground state energy. The singular factor in the nonlinearity significantly limits the smoothness of the ground state and prompts a novel approach to the modulation analysis.
Paper Structure (7 sections, 13 theorems, 151 equations)

This paper contains 7 sections, 13 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Spectral Analysis
  3. The ground state and the linearized operator
  4. Coercivity in $\dot H^1$
  5. Eigenvalues and generalized kernel
  6. Modulation Theory
  7. Convergence of nonscattering, constrained solutions

Key Result

Theorem 1.1

There exist forward-global radial solutions $W^{\pm}$ to NLS with satisfying for some $c>0$ and all $t>0$. The solution $W^{+}$ satisfies while the solution $W^{-}$ is global, satisfies and scatters in $\dot H^{1}(\mathbb{R}^{3})$ as $t\to -\infty$.

Theorems & Definitions (26)

  • Theorem 1.1: Existence of special solutions
  • Remark 1.2
  • Theorem 1.3: Classification of threshold dynamics
  • Remark 1.4
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • proof : Proof of Proposition \ref{['P:orthogonality']}
  • Proposition 2.4
  • proof
  • ...and 16 more