具一般对数非线性项分数阶薛定谔方程的基态解——献给李工宝教授 70 寿辰

数学物理学报 ›› 2025, Vol. 45 ›› Issue (6): 1839-1853.

具一般对数非线性项分数阶薛定谔方程的基态解——献给李工宝教授 70 寿辰

安小明(), 房以宁(), 金正昌()

贵州财经大学数学与统计学院贵阳 550025

收稿日期:2025-03-31 修回日期:2025-05-20 出版日期:2025-12-26 发布日期:2025-11-18
作者简介:安小明,E-mail:xman@mail.gufe.edu.cn
房以宁,E-mail:fangyiningmath@126.com
金正昌,E-mail:3386291801@qq.com
基金资助:
国家自然科学基金(12101150)

Ground State Solution for Fractional Schrödinger Equations with General Logarithmic Nonlinear Terms

Xiaoming An(), Yining Fang(), Zhengchang Jin()

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025

Received:2025-03-31 Revised:2025-05-20 Online:2025-12-26 Published:2025-11-18
Supported by:
NSFC(12101150)

摘要/Abstract

摘要：

该文中, 作者考虑如下具一般对数非线性项的分数阶薛定谔方程

$\begin{equation*} (-\Delta)^s u = u(\log|u|)^{\alpha},\ x\in\mathbb{R}^N, \end{equation*}$

其中 $0<s<1$, $N>2s$, $\alpha\ge 1$ 为常数. 通过观察幂次型薛定谔方程 $(-\Delta)^s u = u(|u|^{\sigma}-1)^{\alpha}$ 在 $\sigma\to 0^+$ 时的收敛现象, 证明该问题在 $(-1)^\alpha = -1$ 时存在一个径向正基态解.

关键词: 分数阶薛定谔方程, 对数非线性项, 幂次, 收敛, 基态解.

Abstract:

In this paper, we consider the following Schrödinger equations with general logarithmic nonlinear terms

$\begin{equation*} (-\Delta)^s u = u(\log|u|)^{\alpha}\ \text{in}\ \mathbb{R}^N, \end{equation*}$

where $0<s<1$, $N>2s$, $\alpha\ge 1$ is a constant. By observing the convergent phenomenon of the power-law Schrödinger equation $(-\Delta)^s u = u(|u|^{\sigma}-1)^{\alpha}$ as $\sigma\to 0^+$, we show that the problem has a positive ground state solution if $(-1)^{\alpha}=-1$.

Key words: fractional Schr?dinger equations, logarithmic nonlinear term, power-law, convergence, ground state solution.

中图分类号:

O175.2

安小明, 房以宁, 金正昌. 具一般对数非线性项分数阶薛定谔方程的基态解——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1839-1853.

Xiaoming An, Yining Fang, Zhengchang Jin. Ground State Solution for Fractional Schrödinger Equations with General Logarithmic Nonlinear Terms[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1839-1853.

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