Nonlinear Schrodinger equation with Coulomb potential
Nonlinear Schrodinger equation with Coulomb potential
复制标题
具有库仑势的非线性薛定谔方程
DOI：
10.1007/s10473-022-0606-x
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发表时间：
2022
影响因子：
1
通讯作者：
zheng jiqiang
中科院分区：
文献类型：
--
作者：
miao changxing;zhang junyong;zheng jiqiang
In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{i}}{\partial _t}u + \Delta u + {K \over {\left| x \right|}}u = \lambda {\left| u \right|^{p - 1}}u$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 < p \le 5\,\,{\rm{on}}\,\,{\mathbb{R}^3}$$\end{document}. Our results reveal the influence of the long range potentialK∣x∣−1on the existence and scattering theories for nonlinear Schrödinger equations. In particular, we prove the global existence when the Coulomb potential is attractive, i.e., whenK> 0, and the scattering theory when the Coulomb potential is repulsive, i.e., whenK≤ 0. 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.
In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{i}}{\partial _t}u + \Delta u + {K \over {\left| x \right|}}u = \lambda {\left| u \right|^{p - 1}}u$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 < p \le 5\,\,{\rm{on}}\,\,{\mathbb{R}^3}$$\end{document}. Our results reveal the influence of the long range potentialK∣x∣−1on the existence and scattering theories for nonlinear Schrödinger equations. In particular, we prove the global existence when the Coulomb potential is attractive, i.e., whenK> 0, and the scattering theory when the Coulomb potential is repulsive, i.e., whenK≤ 0. 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.