Infinitely many small energy solutions for the Schrödinger-Poisson equations with magnetic field

Huiling Niu; Junshan Liu; Jiayin Liu; Jun Zheng; Huiling Niu; Junshan Liu; Jiayin Liu; Jun Zheng

doi:10.3934/math.2026509

AIMS Mathematics

2026, Volume 11, Issue 5: 12397-12413. doi: 10.3934/math.2026509

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Research article

Infinitely many small energy solutions for the Schrödinger-Poisson equations with magnetic field

1.
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, 730070, China
2.
School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China
3.
School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, China
4.
Department of Electrical Engineering, Polytechnique Montreal, P.O. Box 6079, Station Centre-Ville, Montreal, QC, Canada H3T 1J4

Received: 12 March 2026 Revised: 23 April 2026 Accepted: 27 April 2026 Published: 06 May 2026
MSC : 35J10, 35J60, 35B38

In this paper, we consider the following Schrödinger-Poisson equations with magnetic field
$ \begin{equation*} (-i\nabla-A(x))^2 u+\theta(|x|^{-1}*|u|^2)u = f(|u|^2)u, \ u\in H^1(\mathbb{R}^{3}, \mathbb{C}), \end{equation*} $
where $ i $ is the imaginary unit and $ \theta\geq 0 $. The function $ A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3} $ denotes a magnetic potential, and $ V:\mathbb{R}^{3}\rightarrow \mathbb{R} $ is a continuous potential. First, we establish the existence of ground state solutions without imposing the strict monotonicity condition and Ambrosetti-Rabinowitz condition. Then using the dual fountain theorem, we obtain the existence of infinitely many small energy solutions. Our results extend some recent work in the literature.
- Schrödinger-Poisson equation,
- magnetic field,
- ground state solution,
- infinitely many solutions
Citation: Huiling Niu, Junshan Liu, Jiayin Liu, Jun Zheng. Infinitely many small energy solutions for the Schrödinger-Poisson equations with magnetic field[J]. AIMS Mathematics, 2026, 11(5): 12397-12413. doi: 10.3934/math.2026509

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Abstract

In this paper, we consider the following Schrödinger-Poisson equations with magnetic field

$ \begin{equation*} (-i\nabla-A(x))^2 u+\theta(|x|^{-1}*|u|^2)u = f(|u|^2)u, \ u\in H^1(\mathbb{R}^{3}, \mathbb{C}), \end{equation*} $

where $ i $ is the imaginary unit and $ \theta\geq 0 $. The function $ A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3} $ denotes a magnetic potential, and $ V:\mathbb{R}^{3}\rightarrow \mathbb{R} $ is a continuous potential. First, we establish the existence of ground state solutions without imposing the strict monotonicity condition and Ambrosetti-Rabinowitz condition. Then using the dual fountain theorem, we obtain the existence of infinitely many small energy solutions. Our results extend some recent work in the literature.

References

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