A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL

doi:10.1007/s10473-022-0606-x

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (6): 2230-2256.doi: 10.1007/s10473-022-0606-x

A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL

Changxing MIAO¹, Junyong ZHANG^2,3, Jiqiang ZHENG¹

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

收稿日期:2022-07-01 出版日期:2022-12-25 发布日期:2022-12-16
通讯作者: Changxing MIAO, E-mail: miao_changxing@iapcm.ac.cn E-mail:miao_changxing@iapcm.ac.cn
基金资助:
The authors were supported by NSFC (12126409, 12026407, 11831004) and the J. Zheng was also supported by Beijing Natural Science Foundation (1222019).

A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL

Changxing MIAO¹, Junyong ZHANG^2,3, Jiqiang ZHENG¹

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: miao_changxing@iapcm.ac.cn E-mail:miao_changxing@iapcm.ac.cn
Supported by:
The authors were supported by NSFC (12126409, 12026407, 11831004) and the J. Zheng was also supported by Beijing Natural Science Foundation (1222019).

摘要/Abstract

摘要： 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.

关键词: nonlinear Schrödinger equations, long range potential, global well-posedness, blow-up, scattering

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

中图分类号:

35P25

Changxing MIAO, Junyong ZHANG, Jiqiang ZHENG. A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL[J]. 数学物理学报(英文版), 2022, 42(6): 2230-2256.

Changxing MIAO, Junyong ZHANG, Jiqiang ZHENG. A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL[J]. Acta mathematica scientia,Series B, 2022, 42(6): 2230-2256.

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A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL

A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL

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