含位势可饱和 Schrödinger 方程的集中正规化解

数学物理学报 ›› 2026, Vol. 46 ›› Issue (3): 1038-1053.

含位势可饱和 Schrödinger 方程的集中正规化解

龙薇(), 卢淑瑶^*(), 张冬梅()

江西师范大学数学与统计学院南昌 330022

收稿日期:2025-07-15 修回日期:2025-10-21 出版日期:2026-06-26 发布日期:2026-06-16
通讯作者: 卢淑瑶 E-mail:lwhope@jxnu.edu.cn;lusy68@jxnu.edu.cn;497344239@qq.com
作者简介:龙薇, E-mail:lwhope@jxnu.edu.cn;
张冬梅, E-mail:497344239@qq.com
基金资助:
国家自然科学基金(12271223);江西省自然科学基金(20252BAC230002);江西省自然科学基金(20242BAB26001);江西省双高领军人才项目, 江西省研究生创新基金(YC2024-B102)

Concentrated Normalized Solutions of Saturable Schrödinger Equations with the Potential

Wei Long(), Shuyao Lu^*(), Dongmei Zhang()

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022

Received:2025-07-15 Revised:2025-10-21 Online:2026-06-26 Published:2026-06-16
Contact: Shuyao Lu E-mail:lwhope@jxnu.edu.cn;lusy68@jxnu.edu.cn;497344239@qq.com
Supported by:
NSFC(12271223);NSF of Jiangxi Province(20252BAC230002);NSF of Jiangxi Province(20242BAB26001);Double-high talents in Jiangxi Province, Graduate Student Innovation Fund of Jiangxi Provincial Department of Education(YC2024-B102)

摘要/Abstract

摘要：

通过惩罚方法, 该文研究了带有局部型假设位势的可饱和 Schrödinger 方程

$ -\Delta u + V(\varepsilon x)u = \lambda u - \Gamma \frac{u^2}{1 + u^2}u, \quad x \in \mathbb{R}^2 $

正规化解的存在性和集中性. 这里 $\varepsilon$ 表示普朗克常数, $\Gamma < 0$ 是耦合常数.

关键词: 可饱和 Sch?dinger 方程, 惩罚方法, 正规化解.

Abstract:

By penalized methods, we investigate the existence and concentration of normalized solutions to the following saturable Schrödinger equation

$-\Delta u + V(\varepsilon x)u=\lambda u -\Gamma \frac{u^2}{1 + u^2}u, \quad x\in\mathbb{R}^{2},\nonumber$

with the potential satisfying a local type assumption, where $\varepsilon$ denotes the Planck constant, $\Gamma<0$ is a coupling constant.

Key words: saturable Schr?dinger equations, penalized methods, normalized solutions.

中图分类号:

O175.25

龙薇, 卢淑瑶, 张冬梅. 含位势可饱和 Schrödinger 方程的集中正规化解[J]. 数学物理学报, 2026, 46(3): 1038-1053.

Wei Long, Shuyao Lu, Dongmei Zhang. Concentrated Normalized Solutions of Saturable Schrödinger Equations with the Potential[J]. Acta mathematica scientia,Series A, 2026, 46(3): 1038-1053.

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