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

Acta mathematica scientia,Series A ›› 2026, Vol. 46 ›› Issue (4): 1443-1457.

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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)

Abstract

Abstract:

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

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

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

CLC Number:

O175.23

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.

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

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