带混合排异非线性项的薛定谔方程基态的极限性质——献给李工宝教授 70 寿辰

数学物理学报 ›› 2025, Vol. 45 ›› Issue (6): 1942-1960.

带混合排异非线性项的薛定谔方程基态的极限性质——献给李工宝教授 70 寿辰

罗肖^*(), 秦茜

合肥工业大学数学学院合肥 230601

收稿日期:2025-05-28 修回日期:2025-09-03 出版日期:2025-12-26 发布日期:2025-11-18
通讯作者: 罗肖 E-mail:luoxiaohf@163.com
作者简介:秦茜，1351096940@qq.com
基金资助:
国家自然科学基金(12471103);和安徽省自然科学基金(2308085MA05)

Limiting Properties of Ground States for the Schrödinger Equation with Mixed Repulsive Nonlinear Terms

Xiao Luo^*(), Xi Qin

School of Mathematics, Hefei University of Technology, Hefei 230601

Received:2025-05-28 Revised:2025-09-03 Online:2025-12-26 Published:2025-11-18
Contact: Xiao Luo E-mail:luoxiaohf@163.com
Supported by:
NSFC(12471103);Anhui Provincial Natural Science Foundation(2308085MA05)

摘要/Abstract

摘要：

Jeanjean-Lu 在文献 [Calc Var Partial Differential Equations, 2022] 中得到了带混合排异非线性项的薛定谔方程基态的存在性. 该文在此基础上通过分析能量、频率与质量的定量关系, 证明了在恰当的伸缩变换下, 文献 [Calc Var Partial Differential Equations, 2022] 中得到的基态收敛到带单个非线性项的薛定谔方程的基态 (质量衰退时) 或收敛到相应的托马斯-费米方程的基态 (质量爆破时). 特别地, 该文的结论对物理相关的三次-五次薛定谔方程成立.

关键词: 薛定谔方程, 排异非线性项, 基态, 渐进性.

Abstract:

JeanJean-Lu obtained the existence of ground states for the Schrödinger equation with mixed-type nonlinearities in the reference [Calc Var Partial Differential Equations, 2022]. Based on this fact, by analyzing the quantitative relationship between energy, frequency and mass, we prove that under appropriate rescalings, the ground state obtained in [Calc Var Partial Differential Equations, 2022] converges to the ground state of the classical Schrödinger equation with a single nonlinear term (when the mass declines) or converges to the ground state of the corresponding Thomas Fermi equation (when the mass tends to infinity). In particular, our conclusion holds true for the cubic-quintic nonlinear Schrödinger equation.

Key words: Schr?dinger equation, mixed repulsive terms, ground state, limiting property.

中图分类号:

O175.23

罗肖, 秦茜. 带混合排异非线性项的薛定谔方程基态的极限性质——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1942-1960.

Xiao Luo, Xi Qin. Limiting Properties of Ground States for the Schrödinger Equation with Mixed Repulsive Nonlinear Terms[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1942-1960.

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