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Scattering for the Radial Schrödinger Equation with Combined Power-type and Choquard-type Nonlinearities

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Abstract

In this paper, we show the scattering of the radial solution for the nonlinear Schrödinger equation with combined power-type and Choquard-type nonlinearities

$$\rm{i}u_{t}+\Delta u=\lambda_{1}\vert u\vert^{p_{1}-1}u+\lambda_{2}(I_{\alpha}\ast\vert u\vert^{p_{2}})\vert u\vert^{p_{2}-2}u.$$

in the energy space H1(ℝN) for λ1λ2 = −1. We establish a scattering criterion for radial solution together with Morawetz estimate which implies the scattering theory. Results show that the defocusing perturbation terms does not determine the scattering solution in energy space.

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References

  1. Alharbi, M. G., Saanouni, T.: Sharp threshold of global well-posedness vs finite time blow-up for a class of inhomogeneous Choquard equations. J. Math. Phys., 60, 081514 (2019)

    Article  MathSciNet  Google Scholar 

  2. Campos, L.: Scattering of radial solutions to inhomogeneous nonlinear Schrödinger equation. Nonlinear Anal., 202, 112118 (2021)

    Article  Google Scholar 

  3. Cazenave, T.: Semilinear Schrödinger Equations. American Mathematical Society, Providence, RI, 2003

    Book  Google Scholar 

  4. Cazenave, T., Fang, D., Xie, J.: Scattering for the focusing energy-subcritical nonlinear Schrödinger equation. Sci. China Math., 54, 2037–2062 (2011)

    Article  MathSciNet  Google Scholar 

  5. Dodson, B., Murphy, J.: A new proof of scattering below the ground state for the 3d radial focusing cubic NLS. Proc. Amer. Math. Soc., 145, 4859–4867 (2017)

    Article  MathSciNet  Google Scholar 

  6. Dodson, B., Murphy, J.: A new proof of scattering below the ground state for the non-radial focusing NLS. Math. Res. Lett., 25, 1805–1825 (2018)

    Article  MathSciNet  Google Scholar 

  7. Duyckaerts, T., Holmer, J., Roudenko, S.: Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett., 15, 1233–1250 (2008)

    Article  MathSciNet  Google Scholar 

  8. Feng, B., Yuan, X.: On the Cauchy problem for the Schrödinger-Hartree equation. Evol. Equat. and Cont. Theory, 4, 431–445 (2015)

    Article  Google Scholar 

  9. Fochi, D.: Inhomogeneous Strichartz estimate. J. Hyperbolic Differ. Equ., 2, 1–24 (2005)

    Article  MathSciNet  Google Scholar 

  10. Gross, E. P., Meeron, E.: Physics of Many-Particle Systems, Gordon Breach, New York, 1966

    Google Scholar 

  11. Holmer, J., Roudenko, S.: A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Comm. Math. Phys., 282, 435–467 (2008)

    Article  MathSciNet  Google Scholar 

  12. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math., 120, 955–980 (1998)

    Article  MathSciNet  Google Scholar 

  13. Kenig, C., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case. Invent. Math., 166, 645–675 (2006)

    Article  MathSciNet  Google Scholar 

  14. Lewin, M., Rougerie, N.: Derivation of Pekar’s polarons from a microscopic model of quantum crystal. SIAM J. Math. Anal., 45, 1267–1301 (2013)

    Article  MathSciNet  Google Scholar 

  15. Lions, P. L.: The Choquard equation and related queations. Nonlinear Anal., 4, 1063–1072 (1980)

    Article  MathSciNet  Google Scholar 

  16. Miao, C., Xu, G., Zhao, L.: Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial case. J. Func. Anal., 253, 605–627 (2007)

    Article  Google Scholar 

  17. Miao, C., Xu, G., Zhao, L.: The Cauchy problem of the Hartree equation. J. PDE, 21, 22–44 (2008)

    MathSciNet  Google Scholar 

  18. Miao, C., Xu, G., Zhao, L.: The dynamics of the 3D radial NLS with the combined terms. Comm. Math. Phys., 318, 767–808 (2013)

    Article  MathSciNet  Google Scholar 

  19. Miao, C., Xu, G., Zhao, L.: The dynamics of the NLS with the combined terms in five and higher dimensions. In: Some Topics in Harmonic Analysis and Applications, ALM 34, Higher Education Press and International Press, Beijing-Boston, 2015, 265–298

    Google Scholar 

  20. Miao, C., Zhao, T., Zheng, J.: On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold. Cal. Var. PDE, 56, Art. 179, 39 pp. (2017)

  21. Saanouni, T.: Scattering threshold for the focusing Choquard equation. Nonlinear Differ. Equ. Appl., 26, (2019)

  22. Sulem, C., Sulem, P. L.: The Nonlinear Schrödinger Equation, Springer, Berlin, 1999

    Google Scholar 

  23. Tao, T.: On the asymptotic behavior of large radial data for a focsuing non-linear Schrödinger equation. Dyn. Partial Differ. Equ., 1, 1–48 (2004)

    Article  MathSciNet  Google Scholar 

  24. Tao, T., Visan, M., Zhang, X.: The nonlinear Schrödinger equation with combined powertype nonlinearities. Comm. Partial Diff. Equ., 32, 1281–1343 (2007)

    Article  Google Scholar 

  25. Wang, Y., Feng, B.: Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities. Bound. Value Probl., 2019, (2019)

  26. Weinstein, M. I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87, 567–576 (1983)

    Article  Google Scholar 

  27. Xu, C., Zhao, T.: A new proof of scattering theory for the 3D radial focusing energy-critical NLS with combined terms. J. Diff. Equ., 268, 6666–6682 (2019)

    Article  MathSciNet  Google Scholar 

  28. Zheng, J.: Focusing NLS with inverse square potential. J. Math. P., 59, 111502 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the associated editor and anonymous referee for their invaluable comments and suggestions which helped to improve the paper greatly.

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Authors and Affiliations

  1. The Graduate School of China Academy of Engineering Physics, Beijing, 100088, P. R. China

    Ying Wang

  2. School of Mathematics and Statistics, Qinghai Normal University, Xining, 810008, P. R. China

    Cheng Bin Xu

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  1. Ying Wang
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  2. Cheng Bin Xu
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Correspondence to Cheng Bin Xu.

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Wang, Y., Xu, C.B. Scattering for the Radial Schrödinger Equation with Combined Power-type and Choquard-type Nonlinearities. Acta. Math. Sin.-English Ser. 40, 1029–1041 (2024). https://doi.org/10.1007/s10114-023-2570-3

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  • DOI: https://doi.org/10.1007/s10114-023-2570-3

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