带 Hartree 型非线性项的对数 Schrödinger 方程解的存在性——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1505-1512.

带 Hartree 型非线性项的对数 Schrödinger 方程解的存在性——献给邓引斌教授 70 寿辰

陈柯衡¹(), 皮慧荣²^,^*()

¹ 广西大学数学学院南宁 530004
² 广西大学数学学院, 广西应用数学中心 (广西大学) 南宁 530004

收稿日期:2026-01-04 修回日期:2026-04-02 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 皮慧荣 E-mail:945051059@qq.com;huirongpi@gxu.edu.cn
作者简介:陈柯衡,E-mail: 945051059@qq.com
基金资助:
国家自然科学基金(12361020)

The Existence of Solutions for the Logarithmic Schrödinger Equation with Hartree-Type Nonlinearity

Keheng Chen¹(), Huirong Pi²^,^*()

¹ School of Mathematics, Guangxi University, Nanning 530004
² School of Mathematics and Center for Applied Mathematics of Guangxi(Guangxi University), Nanning 530004

Received:2026-01-04 Revised:2026-04-02 Online:2026-08-26 Published:2026-06-10
Contact: Huirong Pi E-mail:945051059@qq.com;huirongpi@gxu.edu.cn
Supported by:
NSFC(12361020)

摘要/Abstract

摘要：

该文研究了带 Hartree 型非线性项的对数 Schrödinger 方程

$ \left\{\begin{array}{l} -\Delta u+V(x) u-\phi u=u \log u^{2}, x \in \mathbb{R}^{3}, \\ -\Delta \phi=u^{2}, \phi \in D^{1,2}\left(\mathbb{R}^{3}\right), \end{array}\right. $

其中 $V(x) \in C\left(\mathbb{R}^{3}\right)$ 是非负位势函数. 应用方向导数与约束极小方法, 证明了在 $V(x)$ 满足不同条件下该方程解的存在性.

关键词: 对数 Schr?dinger 方程, Hartree 型非线性项, Nehari 方法

Abstract:

We consider the following logarithmic Schrödinger equation with Hartree-type nonlinearity

$ \left\{\begin{array}{l} -\Delta u+V(x) u-\phi u=u \log u^{2}, x \in \mathbb{R}^{3}, \\ -\Delta \phi=u^{2}, \phi \in D^{1,2}\left(\mathbb{R}^{3}\right), \end{array}\right. $

where $V(x) \in C\left(\mathbb{R}^{3}\right)$ is a nonnegative potential function．By using direction derivative and constrained minimization method, we prove the existence of solutions under different types of potentials.

Key words: logarithmic Schr?dinger equations, Hartree-type nonlinearity, the Nehari method

中图分类号:

O175.25

陈柯衡, 皮慧荣. 带 Hartree 型非线性项的对数 Schrödinger 方程解的存在性——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1505-1512.

Keheng Chen, Huirong Pi. The Existence of Solutions for the Logarithmic Schrödinger Equation with Hartree-Type Nonlinearity[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1505-1512.

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