数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1428-1442.

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具有不定位势的基尔霍夫型拟线性薛定谔-泊松系统的非平凡解——献给邓引斌教授 70 寿辰

黄古珍1(), 王莉1,*(), 汪继秀2()   

  1. 1 华东交通大学理学院 南昌 330013
    2 江汉大学人工智能学院 武汉 430074
  • 收稿日期:2025-12-29 修回日期:2026-02-05 出版日期:2026-08-26 发布日期:2026-06-10
  • 通讯作者: 王莉 E-mail:huangguzhen0428@163.com;wangli.423@163.com;wangjixiu127@aliyun.com
  • 作者简介:黄古珍,E-mail: huangguzhen0428@163.com;
    汪继秀,E-mail: wangjixiu127@aliyun.com
  • 基金资助:
    国家自然科学基金(12161038);江西省自然科学基金(20232BAB201009);江西省教育厅科学技术项目(GJJ2400901)

Nontrivial Solution for the Kirchhoff Type Quasilinear Schrödinger-Poisson Systems with Indefinite Potentials

Guzhen Huang1(), Li Wang1,*(), Jixiu Wang2()   

  1. 1 School of Science, East China Jiaotong University, Nanchang 330013
    2 School of Artificial Intelligence, Jianghan University, Wuhan 430074
  • Received:2025-12-29 Revised:2026-02-05 Online:2026-08-26 Published:2026-06-10
  • Contact: Li Wang E-mail:huangguzhen0428@163.com;wangli.423@163.com;wangjixiu127@aliyun.com
  • Supported by:
    NSFC(12161038);Jiangxi Provincial Natural Science Foundation(20232BAB201009);Science and Technology Project of Education Department of Jiangxi Province(GJJ2400901)

摘要:

该文中研究了如下基尔霍夫型拟线性薛定谔-泊松系统

$\begin{align*} \begin{cases} -\left(a+b\int_{\mathbf{R}^{3}}|\nabla u|^{2}\mathrm{d} x\right)\Delta u + V(x)u + \phi u = f(x, u), & x \in \mathbf{R}^{3}, \\ -\Delta\phi - \varepsilon^{4}\Delta_{4}\phi = u^{2}, & x \in \mathbf{R}^{3}, \end{cases} \end{align*}$

其中位势函数 $V$ 为不定位势, 这使得对应的薛定谔算子 $ -\Delta + V $ 存在有限维的负空间. 通过应用莫尔斯理论, 作者证明了该系统非平凡解的存在性, 还分别讨论了 $\varepsilon\to 0$ 与 $ b \to 0$ 时解的渐近行为.

关键词: 拟线性薛定谔-泊松系统, 局部环绕, 莫尔斯理论

Abstract:

In this paper, we investigate the Kirchhoff type Quasilinear Schrödinger-Poisson systems:

$\begin{align*} \begin{cases} -\left(a+b\int_{\mathbf{R}^{3}}|\nabla u|^{2}\mathrm{d} x\right)\Delta u + V(x)u + \phi u = f(x, u), & x \in \mathbf{R}^{3}, \\ -\Delta\phi - \varepsilon^{4}\Delta_{4}\phi = u^{2}, & x \in \mathbf{R}^{3}, \end{cases} \end{align*}$

where the potential $V$ is indefinite, leading the Schrödinger operator $ -\Delta + V $ exhibit a finite-dimensional negative space. By Morse theory, we prove the existence of nontrivial solutions to this system. It also discusses the asymptotic behavior of the solution as $\varepsilon \to 0$ and $b \to 0$ separately.

Key words: quasilinear Schr?inger-Poisson systems, local linking, morse theory

中图分类号: 

  • O175.23