含对数型非线性项的拟线性薛定谔方程的驻波解——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1572-1584.

含对数型非线性项的拟线性薛定谔方程的驻波解——献给邓引斌教授 70 寿辰

金庆飞()

江汉大学人工智能学院武汉 430056

收稿日期:2026-02-12 修回日期:2026-03-16 出版日期:2026-08-26 发布日期:2026-06-10
作者简介:金庆飞,E-mail: jinqingfei@jhun.edu.cn
基金资助:
江汉大学科研启动基金(06050001)

Standing Wave Solutions of the Quasilinear Schrödinger Equation with Logarithmic Nonlinear Terms

Qingfei Jin()

School of Artificial Intelligence, Jianghan University, Wuhan 430056

Received:2026-02-12 Revised:2026-03-16 Online:2026-08-26 Published:2026-06-10
Supported by:
Research Startup Foundation of Jianghan University(06050001)

摘要/Abstract

摘要：

该文研究一类带参数的含对数型非线性项的拟线性薛定谔方程

$ -\Delta u + V(x)u + \frac{\kappa}{2} [\Delta |u|^2 ]u = u\log (1 + |u|^2), \quad x\in \mathbb{R}^N $

的非平凡经典解的存在性, 其中 $N\geq 3$, $\kappa >0$ 为参数, $V:\mathbb{R}^N\to \mathbb{R}$ 为连续函数. 该模型在等离子体物理和非线性光学中具有重要意义. 结合变分方法和扰动技巧, 证明了当参数 $\kappa$ 充分小时, 该方程存在非平凡解, 并建立了解的 $L^{\infty}$ 估计.

关键词: 拟线性薛定谔方程, 对数非线性项, 山路定理, 驻波, 变分方法

Abstract:

This study investigates the existence of non-trivial classical solutions for a class of parameterized quasilinear Schrödinger equations containing logarithmic nonlinear terms:

$ -\Delta u + V(x)u + \frac{\kappa}{2} [\Delta |u|^2 ]u = u\log (1 + |u|^2), \quad x\in \mathbb{R}^N $

where $N \geq 3$, $\kappa > 0$ is a parameter, and $V:\mathbb{R}^{N} \rightarrow \mathbb{R}$ is a continuous function. The model holds significant importance in plasma physics and nonlinear optics. By combining variational methods and perturbation techniques, we demonstrate that non-trivial solutions exist for sufficiently small parameters $\kappa$, and establish $L^{\infty}$ estimates for these solutions.

Key words: quasilinear Schr?dinger equation, logarithmic nonlinear term, mountain pass theorem, standing wave, variational method

中图分类号:

O175.23

金庆飞. 含对数型非线性项的拟线性薛定谔方程的驻波解——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1572-1584.

Qingfei Jin. Standing Wave Solutions of the Quasilinear Schrödinger Equation with Logarithmic Nonlinear Terms[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1572-1584.

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