Dynamics of Partial Differential Equations

Volume 20 (2023)

Number 1

Dynamics of subcritical threshold solutions for energy-critical NLS

Pages: 37 – 72

DOI: https://dx.doi.org/10.4310/DPDE.2023.v20.n1.a3


Qingtang Su (Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, China; and Morningside Center of Mathematics, Beijing, China)

Zehua Zhao (Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China; and MIIT Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing, China)


In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on $\mathbb{R}^d \, (d \geq 5)$ with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in [19] for $d = 3, 4, 5$ and later by D. Li and X. Zhang in [25] for $d \geq 6$. We generalize the conclusion for the subcritical threshold solutions by removing the radial assumption for $d \geq 5$. A key step is to show exponential convergence to the ground state $W(x)$ up to symmetries if the scattering phenomenon does not occur. Remarkably, an interaction Morawetz-type estimate is applied.


focusing NLS, energy-critical, ground state, threshold solution, interaction Morawetz estimate

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35R01, 37Kxx, 37L50.

Z. Zhao was supported by the NSF grant of China (No. 12101046, 12271032), and by the Beijing Institute of Technology Research Fund Program for Young Scholars.

Received 5 November 2020

Published 23 December 2022