周期位势下 Schrödinger-Bopp-Podolsky 系统非平凡解的存在性——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1443-1457.

周期位势下 Schrödinger-Bopp-Podolsky 系统非平凡解的存在性——献给邓引斌教授 70 寿辰

蔡紫杏(), 王莉^*(), 杨雨晨()

华东交通大学理学院南昌 330013

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:caizixing1229@163.com;wangli.423@163.com;yangyuchen0821@163.com
作者简介:蔡紫杏,E-mail: caizixing1229@163.com;
杨雨晨,E-mail: yangyuchen0821@163.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)

Existence of Nontrivial Solutions for a Schrödinger-Bopp-Podolsky System in $\mathbf{R}^3$ with Periodic Potentials

Zixing Cai(), Li Wang^*(), Yuchen Yang()

School of Science, East China Jiaotong University, Nanchang 330013

Received:2025-12-29 Revised:2026-02-05 Online:2026-08-26 Published:2026-06-10
Contact: Li Wang E-mail:caizixing1229@163.com;wangli.423@163.com;yangyuchen0821@163.com
Supported by:
NSFC(12161038);Jiangxi Provincial Natural Science Foundation(20232BAB201009);Science and Technology Project of Education Department of Jiangxi Province(GJJ2400901)

摘要/Abstract

摘要：

该文研究如下定义在 $\mathbf{R}^3$ 上的 Schrödinger-Bopp-Podolsky 系统

$\left\{\begin{aligned}&-\left( a+b\int_{\mathbf{R}^3}|\nabla u|^2\,\mathrm{d}x\right) \Delta u + V(x)u+\lambda\phi u = f(x,u), && x \in \mathbf{R}^3, \\&-\Delta \phi+d^2\Delta ^{2}\phi = \lambda u^2, && x \in \mathbf{R}^3,\end{aligned}\right.$

其中 $a,$ $b>0$ 为常数, $\lambda,$ $d$ 为正参数; 位势函数 $V(x)$ 为连续的周期位势, 并具有正下界; 非线性项 $f(x,t)\in C(\mathbf{R}^3\times\mathbf{R},\mathbf{R})$ 是关于 $x$ 的周期函数. 在 $f$ 满足一定条件下, 借助变分方法与截断技巧, 作者在固定 $d$ 且 $\lambda$ 充分小时获得一个非平凡解, 并进一步给出解的渐近行为, 所得结果推广并改进了近期相关文献中的结论.

关键词: Schr?dinger-Bopp-Podolsky 系统, 周期位势, 截断技巧

Abstract:

In this paper, we investigate the following Schrödinger-Bopp-Podolsky system in $\mathbf{R}^3$:

where $a,$ $b>0$ are constants, $\lambda,$ $d$ are positive parameters, $V(x)$ is a continuous and periodic potential function with positive infimum, $f(x,t)\in C(\mathbf{R}^3\times\mathbf{R},\mathbf{R})$ is periodic in $x.$ Under $f(x,t)$ satisfying some superquadratic growth conditions with respect to $t,$ by combining variational methods with a truncation technique, we obtain one nontrivial solution for $\lambda$ small enough and $ d$ fixed. The asymptotic behavior of this solution is also discussed in this paper. Our results generalize and improve some recent results in the literature.

Key words: Schr?dinger-Bopp-Podolsky system, periodic potential, truncation technique

中图分类号:

O175.23

蔡紫杏, 王莉, 杨雨晨. 周期位势下 Schrödinger-Bopp-Podolsky 系统非平凡解的存在性——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1443-1457.

Zixing Cai, Li Wang, Yuchen Yang. Existence of Nontrivial Solutions for a Schrödinger-Bopp-Podolsky System in $\mathbf{R}^3$ with Periodic Potentials[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1443-1457.

参考文献 17

[1]	Bopp F. Eine lineare Theorie des Elektrons. Ann Phys, 1940, 430: 345-384 doi: 10.1002/andp.v430:5
[2]	Podolsky B. A generalized electrodynamics. I Nonquantum, Phys Rev, 1942, 62: 68-71
[3]	Aberqi A, Bennouna J, Benslimane O, et al. On $p(z)$-Laplacian system involving critical nonlinearities. J Funct Spaces, 2022, 2022(1): Art 6855771
[4]	Alghandi A M, Gala S, Ragusa M A. Improved regularity criterion for the 3D Navier-Stokes equations via the gradient of one velocity component. AIP Conf Proc, 2022, 2425(1): Art 280003
[5]	Farid M, Ali R, Kazmi K R. Inertial iterative method for a generalized mixed equilibrium, variational inequality and a fixed point problems for a family of quasi-nonexpansive mappings. Filomat, 2023, 37: 6133-6150 doi: 10.2298/FIL2318133F
[6]	Frenkel F. 4/3 problem in classical electrodynamics. Phys Rev E, 1996, 54: 5859-5862 pmid: 9965782
[7]	Kirchhoff G. Mechanik. Leipzig: Teubner, 1883
[8]	Lions J L. On some questions in boundary value problems of mathematical physics. North-Holland Mathematics Studies, 1978, 30: 284-346
[9]	Li G, Ye H. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbf{R}^3$. J Differ Equ, 2014, 257(2): 566-600 doi: 10.1016/j.jde.2014.04.011
[10]	Perera K, Zhang Z. Nontrivial solutions of Kirchhoff-type problems via the Yang index. J Differ Equ, 2006, 221(1): 246-255 doi: 10.1016/j.jde.2005.03.006
[11]	Ma T F, Rivera J E M. Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl Math Lett, 2003, 16(2): 243-248 doi: 10.1016/S0893-9659(03)80038-1
[12]	He X, Zou W. Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal, 2009, 70(3): 1407-1414 doi: 10.1016/j.na.2008.02.021
[13]	Ma X, He X. Nontrivial solutions for Kirchhoff equations with periodic potentials, Electronic Journal of Differential Equations, 2016, 2016(102): 1-22
[14]	D'Avenia P, Siciliano G. Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: Solutions in the electrostatic case. J Differ Equ, 2019, 267: 1025-1065 doi: 10.1016/j.jde.2019.02.001
[15]	Li Y, Rădulescu V D, Zhang B. Critical planar Schrödinger-Poisson equations: Existence, multiplicity and concentration. Mathematische Zeitschrift, 2024, 307: Art 43
[16]	Damian H M S, Siciliano G. Critical Schrödinger-Bopp-Podolsky systems: Solutions in the semiclassical limit. Calc Var, 2024, 63: Art 155
[17]	Lions P L. The concentration-compactness principle in the calculus of variations. The limit case, Part 2. Rev Mat Iberoam, 1985, 1(2): 45-121