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On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold

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Abstract

In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schrödinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension. This extends the results in Miao et al. (Commun Math Phys 318(3):767–808, 2013, The dynamics of the NLS with the combined terms in five and higher dimensions. Some topics in harmonic analysis and applications, advanced lectures in mathematics, ALM34, Higher Education Press, Beijing, pp 265–298, 2015) to four dimension without radial assumption and the proof of scattering is based on the interaction Morawetz estimates developed in Dodson (Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension \(d =4\) for initial data below a ground state threshold, arXiv:1409.1950), the main ingredients of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

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Acknowledgements

The authors thank the referee and the associated editor for their invaluable comments and suggestions which helped improve the paper greatly. This work was supported in part by the National Natural Science Foundation of China under Grants 11671047.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, 100088, China

    Changxing Miao

  2. The Graduate School of China Academy of Engineering Physics, P. O. Box 2101, Beijing, 100088, China

    Tengfei Zhao

  3. Université Nice Sophia-Antipolis, 06108, Nice Cedex 02, France

    Jiqiang Zheng

Authors
  1. Changxing Miao
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  2. Tengfei Zhao
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  3. Jiqiang Zheng
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Correspondence to Changxing Miao.

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Communicated by F. H. Lin.

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Miao, C., Zhao, T. & Zheng, J. On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold. Calc. Var. 56, 179 (2017). https://doi.org/10.1007/s00526-017-1264-z

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  • DOI: https://doi.org/10.1007/s00526-017-1264-z

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