一类带有吸引库伦势的 Schrödinger 方程正规化解的多重性——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1344-1359.

一类带有吸引库伦势的 Schrödinger 方程正规化解的多重性——献给邓引斌教授 70 寿辰

陆璐()

中南财经政法大学统计与数学学院武汉 430073

收稿日期:2025-12-15 修回日期:2026-01-20 出版日期:2026-08-26 发布日期:2026-06-10
作者简介:陆璐,E-mail: lulu@zuel.edu.cn

Multiple of Normalized Solutions of a Class of Schrödinger Equations with Attractive Coulomb Potential

Lu Lu()

School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073

Received:2025-12-15 Revised:2026-01-20 Online:2026-08-26 Published:2026-06-10

摘要/Abstract

摘要：

该文主要关注如下带有吸引库伦势的 Schrödinger 方程

$-\Delta u -|x|^{-1}u-|u|^{p-2}u-\lambda u=0,\,\,\,\,\ x\in\mathbb{R}^3,$

在正规化约束下

$\int_{\mathbb{R}^3} u^2(x){\rm d}x=c$

的解, 其中 $p\in(\frac{10}{3},6), \lambda\in\mathbb{R}$. 作者证明当质量 $c$ 较小时, 上述方程存在基态解, 且对应于相关能量泛函的局部极小值点. 同时, 作者也得到了方程激发态解的存在性, 并且进一步证明了激发态解位于能量泛函的一个山路水平. 最后, 通过极小极大方法, 作者建立了无穷多高能量解的存在性.

关键词: Schr?dinger方程, 库伦势, 正规化解

Abstract:

In this paper, we focus on the solutions to the Schrödinger equation with attractive Coulomb potential

$-\Delta u -|x|^{-1}u-|u|^{p-2}u-\lambda u=0,\,\,\,\,\ x\in\mathbb{R}^3,$

under the normalized constriant

$\int_{\mathbb{R}^3} u^2(x){\rm d}x=c$

where $p\in(\frac{10}{3},6), \lambda\in\mathbb{R}$. We show that for small mass, the ground states exist and correspond to local minima of the associated energy functional. The existence of the excited states is also obtained. We next prove that the excited states are located at a mountain-pass level of energy functional. Finally, the existence of infinitely many high energy solutions is established by using a minimax procedure.

Key words: Schr?dinger equation, coulomb potential, normalized solutions

中图分类号:

O175.23

陆璐. 一类带有吸引库伦势的 Schrödinger 方程正规化解的多重性——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1344-1359.

Lu Lu. Multiple of Normalized Solutions of a Class of Schrödinger Equations with Attractive Coulomb Potential[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1344-1359.

参考文献 20

[1]	Bartsch T, De Valeriola S. Normalized solutions of nonlinear Schrödinger equations. Arch Math, 2013, 100(1): 75-83 doi: 10.1007/s00013-012-0468-x
[2]	Benguria R, Jeanneret L. Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on $\mathbb{R}^3$. Commun Math Phys, 1986, 104(2): 291-306 doi: 10.1007/BF01211596
[3]	Berestycki H, Lions P L. Nonlinear scalar field equations II. Arch Rational Mech Anal, 1983, 82: 347-375 doi: 10.1007/BF00250556
[4]	Dinh V D. On nonlinear Schrödinger equations with attractive inverse-power potentials. Topol Methods Nonlinear Anal, 2021, 57(2): 489-523
[5]	Ghoussoub N. Duality and Perturbation Methods in Critical Point Theory. Cambridge: Cambridge Univ Press, 1993
[6]	Hardy G, Littlewood J E, Polya G. Inequalities. Cambridge: Cambridge Univ Press, 1934
[7]	Jeanjean L. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal, 1997, 28: 1633-1659 doi: 10.1016/S0362-546X(96)00021-1
[8]	Jeanjean L, Jendrej J, Le T T. Orbital stability of ground states for a Sobolev critical Schrödinger equation. J Math Pures Appl, 2022, 164: 158-179 doi: 10.1016/j.matpur.2022.06.005
[9]	Jeanjean L, Le T T. Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation. J Differential Equations, 2021, 303: 277-325 doi: 10.1016/j.jde.2021.09.022
[10]	Jeanjean L, Le T T. Multiple normalized solutions for a Sobolev critical Schrödinger equation. Math Ann, 2022, 384: 101-134 doi: 10.1007/s00208-021-02228-0
[11]	Jeblick M, Leopold N, Pickl P. Derivation of the time dependent Gross-Pitaevskii equation in two dimensions. Commun Math Phys, 2019, 372: 1-69 doi: 10.1007/s00220-019-03599-x pmid: 32675823
[12]	Kirkpatrick K, Schlein B, Staffilani G. Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics. Am J Math, 2011, 133: 91-130 doi: 10.1353/ajm.2011.0004
[13]	Li X, Zhao J. Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential. Comput Math Appl, 2020, 79: 303-316 doi: 10.1016/j.camwa.2019.06.030
[14]	Li Y, Yao F. Derivation of the nonlinear Schrödinger equation with a general nonlinearity and Gross-Pitaevskii hierarchy in one and two dimensions. J Math Phys, 2021, 62: 021505 doi: 10.1063/5.0035676
[15]	Lieb E H, Loss M. Analysis. Providence, RI: American Mathematical Society, 2001
[16]	Luo X, Wang Q F. Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in $\mathbb{R}^3$. Nonlinear Anal RWA, 2017, 33: 19-32 doi: 10.1016/j.nonrwa.2016.06.001
[17]	Miao C, Zhang J, Zheng J. A nonlinear Schrödinger equation with Coulomb potential. Acta Math Sci, 2022, 42B(6): 2230-2256
[18]	Struwe M. Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Berlin: Springer-Verlag, 1996
[19]	Taylor M E. Partial Differential Equations II. Qualitative Studies of Linear Equations. New York: Springer, 2011
[20]	Weinstein M I. Nonlinear Schrödinger equations and sharp interpolations estimates. Commun Math Phys, 1983, 87: 567-576 doi: 10.1007/BF01208265