Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (6): 2230-2256.doi: 10.1007/s10473-022-0606-x

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Changxing MIAO1, Junyong ZHANG2,3, Jiqiang ZHENG1   

  1. 1. Institute of Applied Physics and Computational Mathematics, Beijing 100088, China;
    2. Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China;
    3. Department of Mathematics, Cardiff University, UK
  • Received:2022-07-01 Online:2022-12-25 Published:2022-12-16
  • Contact: Changxing MIAO, E-mail:
  • Supported by:
    The authors were supported by NSFC (12126409, 12026407, 11831004) and the J. Zheng was also supported by Beijing Natural Science Foundation (1222019).

Abstract: In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential i$?_tu+\Delta u+\frac{K}{|x|}u=\lambda|u|^{p-1}u$ with 1<p≤5 on $\mathbb{R}^3$. Our results reveal the influence of the long range potential $K|x|^{-1}$ on the existence and scattering theories for nonlinear Schrödinger equations. In particular, we prove the global existence when the Coulomb potential is attractive, i.e., when $K>0$, and the scattering theory when the Coulomb potential is repulsive, i.e., when $K\leq0$. The argument is based on the newly-established interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.

Key words: nonlinear Schrödinger equations, long range potential, global well-posedness, blow-up, scattering

CLC Number: 

  • 35P25