Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 4

On the energy-critical fractional Schrödinger equation in the radial case

Pages: 265 – 282

DOI: https://dx.doi.org/10.4310/DPDE.2018.v15.n4.a2


Zihua Guo (School of Mathematical Sciences, Monash University, Melbourne, Victoria, Australia)

Yannick Sire (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Yuzhao Wang (Department of Mathematics and Physics, North China Electric Power University, Beijing, China; and School of Mathematics, University of Birmingham, United Kingdom)

Lifeng Zhao (School of Mathematical Sciences, University of Science and Technology of China, Hefei, China)


We consider the Cauchy problem for the energy-critical nonlinear Schrödinger equation with fractional Laplacian (fNLS) in the radial case. We obtain global well-posedness and scattering in the energy space in the defocusing case, and in the focusing case with energy below the ground state. The main feature of the present work is the nonlocality of the operator. This does not allow us to use standard computations for the rigidity part of the theorem. Instead we develop a commutator argument which has its own interest for problems with nonlocal operators.


nonlinear wave equation, nonlinear Schrödinger equation

2010 Mathematics Subject Classification

35L70, 35Q55

This work was supported by Australian Research Council Discovery Project (Grant No. DP170101060). L. Zhao is supported by grant NSFC of China no. 11771415. Y. Wang is supported by NSFC of China, no. 11771140. Y. Sire is supported by a Simons fellowship.

Received 27 November 2017

Published 5 December 2018