English

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

Analysis of PDEs 2026-01-12 v1

Abstract

We study the energy-critical 3d3d cubic inhomogeneous NLS equation itu+Δu+x1u2u=0i\partial_t u + \Delta u + |x|^{-1}|u|^2 u=0. In this work, we prove the existence of special solutions W±W^\pm with energy equal to that of the ground state WW and use these solutions to characterize the behavior of solutions at the ground state energy. The singular factor x1|x|^{-1} in the nonlinearity significantly limits the smoothness of the ground state and prompts a novel approach to the modulation analysis.

Keywords

Cite

@article{arxiv.2601.05349,
  title  = {Threshold solutions for the $3d$ cubic INLS: the energy-critical case},
  author = {Luccas Campos and Luiz Gustavo Farah and Jason Murphy},
  journal= {arXiv preprint arXiv:2601.05349},
  year   = {2026}
}