GROUND STATE TO A NEW CLASS OF KIRCHHOFF-TYPE EQUATION WITH DOUBLY CRITICAL HARTREE-TYPE NONLINEARITIES

Li Zhou; Chuanxi Zhu; Shufen Liu

doi:10.11948/20250159

2026 Volume 16 Issue 2

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Li Zhou, Chuanxi Zhu, Shufen Liu. GROUND STATE TO A NEW CLASS OF KIRCHHOFF-TYPE EQUATION WITH DOUBLY CRITICAL HARTREE-TYPE NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 1095-1108. doi: 10.11948/20250159

Citation:

Li Zhou, Chuanxi Zhu, Shufen Liu. GROUND STATE TO A NEW CLASS OF KIRCHHOFF-TYPE EQUATION WITH DOUBLY CRITICAL HARTREE-TYPE NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 1095-1108. doi: 10.11948/20250159

GROUND STATE TO A NEW CLASS OF KIRCHHOFF-TYPE EQUATION WITH DOUBLY CRITICAL HARTREE-TYPE NONLINEARITIES

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: 397621669@qq.com(L. Zhou); Email: 2573627764@qq.com(S. Liu)
Corresponding author: Email: 2544537705@qq.com(C. Zhu)
Fund Project: The authors were supported by Scientific research start-up fund (Grant No. F701108M03), National Natural Science Foundation of China (11771198, 11901276) and Science and Technology project of Education Department of Jiangxi Province (Grant No. GJJ2403004)

Abstract

Abstract

In this paper, we consider the following class of Kirchhoff-type equation with doubly critical Hartree-type nonlinearities
$ \left\{ \begin{align}& -(a+b\int _{\mathbb{R}^N}|\nabla u|^2\text{d}x)\Delta u+V(x)u=(I_{\alpha}*|u|^p)|u|^{p-2}u+(I_{\alpha}*|u|^q)|u|^{q-2}u,\,\, \text{in}\,\,\mathbb{R}^N,\\& u\,\in\,H^1(\mathbb{R}^N), \end{align} \right. $
where $ a>0 $, $ b\geq0 $, $ N\geq 3 $, $ \alpha\,\in\,(N-2,N) $, $ p=\frac{N+\alpha }{N-2} $, $ q=\frac{N+\alpha }{N} $, $ V:\,\mathbb{R}^N \rightarrow \mathbb{R} $ is a potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\,\in\,(N-2,N) $. Under certain assumptions on potential function $ V(x) $, we prove that the equation has at least a ground state solution by variational methods.
- Kirchhoff equation /
- ground state solutions /
- Pohozaev identity /
- Nehari manifold
MSC: 35J60, 35J35, 35A15

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References

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