DCDS
The energy-critical NLS with inverse-square potential
Rowan Killip Changxing Miao Monica Visan Junyong Zhang Jiqiang Zheng
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 3831-3866 doi: 10.3934/dcds.2017162

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δ u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a > - \frac{1}{4} + \frac{1}{{25}}$. We also carry out the variational analysis needed to treat the focusing case.

keywords: Nonlinear Schröodinger equation scattering inverse-square potential concentration compactness
DCDS
Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data
Chenmin Sun Hua Wang Xiaohua Yao Jiqiang Zheng
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 2207-2228 doi: 10.3934/dcds.2018091

The aim of this paper is to adapt the strategy in [8] [ See, B. Dodson, J. Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for certain focusing fractional NLS. The main ingredient is to apply the fractional virial identity proved in [3] [ See, T. Boulenger, D. Himmelsbach, E. Lenzmann, Blow up for fractional NLS, J. Func. Anal, 271(2016), 2569-2603 ] to exclude the concentration of mass near the origin.

keywords: Fractional Schrödinger equation scattering Morawetz estimate

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