四阶薛定谔方程归一化解存在性的另一个证明——献给李工宝教授 70 寿辰

数学物理学报 ›› 2025, Vol. 45 ›› Issue (6): 1928-1941.

四阶薛定谔方程归一化解存在性的另一个证明——献给李工宝教授 70 寿辰

吴梦霞¹, 杨涛¹^,^*(), 张华²

¹浙江师范大学数学科学学院浙江金华 321004
²郧西县第一中学湖北十堰 442600

收稿日期:2025-05-14 修回日期:2025-07-18 出版日期:2025-12-26 发布日期:2025-11-18
通讯作者: 杨涛 E-mail:yangtao@zjnu.edu.cn
基金资助:
国家自然科学基金(12201564);浙江师范大学科研启动金(YS304221948)

Another Proof on the Existence of Normalized Solution to a Fourth-Order Schrödinger Equation

Mengxia Wu¹, Tao Yang¹^,^*(), Hua Zhang²

¹School of Mathematical Sciences, Zhejiang Normal University, Zhejiang Jinhua 321004
²Yunxi No.1 Senior High School, Hubei Shiyan 442600

Received:2025-05-14 Revised:2025-07-18 Online:2025-12-26 Published:2025-11-18
Contact: Tao Yang E-mail:yangtao@zjnu.edu.cn
Supported by:
NSFC(12201564);Scientific Research Fund of Zhejiang Normal University(YS304221948)

摘要/Abstract

摘要：

该文研究了具有正二阶色散系数的四阶薛定谔方程的归一化解的存在性和渐近性. 在质量超临界情形, 通过引入两类局部极小化问题并且证明其等价, 回避了局部化限制半径对质量的依赖性, 证明了相应极小化序列的紧性, 得到了方程基态解的存在性. 进一步, 借助细致的能量估计和分析, 也给出了基态解和拉格朗日乘子在参数趋近于零时的渐近性质. 该文去掉了文献 [7] (Sci China Math, 2023, 66: 1237-1262) 中的径向对称性条件, 给出了比文献 [8] (J Differential Equations, 2022, 330: 1-65) 更简洁的证明方法.

关键词: 四阶薛定谔方程, 归一化解, 局部极小化方法.

Abstract:

In this paper, we consider the existence and asymptotic properties of normalized solutions to a fourth-order Schrödinger equation with a positive second-order dispersion coefficient. In the mass supercritical regime, we study two types of local minimization problems and prove their equivalence in order to avoid the dependence of mass with respect to the locally constraint radius. Then, we prove the compactness of the corresponding minimizing sequences and the existence of ground states. Furthermore, by utilizing subtle energy estimates and analysis, we derive the asymptotic behavior of the ground state and the Lagrange multiplier as the parameter vanishes. This paper removes the radial symmetry condition in (Sci China Math, 2023, 66: 1237--1262), and provides an alternative but more transparent proof than that of (J Differential Equations, 2022, 330: 1--65).

Key words: fourth-order Schr?dinger equation, normalized solution, local minimization method.

中图分类号:

O175.23

吴梦霞, 杨涛, 张华. 四阶薛定谔方程归一化解存在性的另一个证明——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1928-1941.

Mengxia Wu, Tao Yang, Hua Zhang. Another Proof on the Existence of Normalized Solution to a Fourth-Order Schrödinger Equation[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1928-1941.

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