EXISTENCE AND ASYMPTOTIC BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF MAGNETIC KIRCHHOFF CHOQUARD TYPE EQUATION WITH A STEEP POTENTIAL WELL

Li Zhou; Chuanxi Zhu; Shufen Liu

doi:10.11948/20230226

2024 Volume 14 Issue 1

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Li Zhou, Chuanxi Zhu, Shufen Liu. EXISTENCE AND ASYMPTOTIC BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF MAGNETIC KIRCHHOFF CHOQUARD TYPE EQUATION WITH A STEEP POTENTIAL WELL[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 379-391. doi: 10.11948/20230226

Citation:

Li Zhou, Chuanxi Zhu, Shufen Liu. EXISTENCE AND ASYMPTOTIC BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF MAGNETIC KIRCHHOFF CHOQUARD TYPE EQUATION WITH A STEEP POTENTIAL WELL[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 379-391. doi: 10.11948/20230226

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF MAGNETIC KIRCHHOFF CHOQUARD TYPE EQUATION WITH A STEEP POTENTIAL WELL

1.
Department of Mathematics, Zhejiang University of Science & Technology, Hangzhou, Zhejiang 310023, China
2.
School of Mathematics, Dalian University of Technology, Dalian, Liaoning 116024, China
3.
Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China
4.
Department of Basic Discipline, Nanchang JiaoTong Institute, Nanchang, Jiangxi 330031, China

Author Bio: Email: zhouli8869@163.com(L. Zhou); Email: 2573627764@qq.com(C. Zhu)
Corresponding author: Email: 397621669@qq.com(S. Liu)
Fund Project: The authors were supported by Scientific research start-up fund (Grant No. F701108M03), National Natural Science Foundation of China (11771198, 11901276) and School level teaching research & reform project (Grant No. 2023-jg51)

Abstract

Abstract

In this paper, we consider the following nonlinear magnetic Kirchhoff Choquard type equation
$ \begin{align*} &[a+b\int _{ \mathbb{R}^N}(|\nabla_A u|^2+\lambda V(x)|u|^2)\text{d}x](-\Delta_A u+\lambda V(x)u)\\=&(I_{\alpha}*F(|u|))\frac{f(|u|)}{|u|}u, \, \, \text{in}\, \, \mathbb{R}^N, \end{align*} $
where $ u: \mathbb{R}^N\rightarrow \mathbb{C} $, $ A: \mathbb{R}^N\rightarrow \mathbb{R}^N $ is a vector potential, $ N\geq 3 $, $ a>0 $, $ b>0 $, $ \alpha\, \in\, (N-2, N] $, $ V:\, \mathbb{R}^N \rightarrow \mathbb{R} $ is a scalar potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\, \in\, (N-2, N] $. Under certain assumptions on $ A(x) $, $ V(x) $ and $ f(t) $, we prove that the equation has at least one ground state solution by variational methods and investigate the asymptotic behavior of solutions.
- Magnetic Laplace operator /
- ground state solutions /
- Nehari manifold /
- asymptotic behavior
MSC: 35J60, 35J20

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References

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