MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION

MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION

MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION

MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION

MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION

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doi:10.1007/s10473-025-0218-3

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (2): 602-614.doi: 10.1007/s10473-025-0218-3

Peng Chen^1,2,*, Longjiang Gu³, Yan Wu⁴

1. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China;
2. College of Science, China Three Gorges University, Yichang 443002, China;
3. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China;
4. School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, China

收稿日期:2024-02-28 出版日期:2025-03-25 发布日期:2025-05-08

Peng Chen^1,2,*, Longjiang Gu³, Yan Wu⁴

1. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China;
2. College of Science, China Three Gorges University, Yichang 443002, China;
3. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China;
4. School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, China

Received:2024-02-28 Online:2025-03-25 Published:2025-05-08
Contact: *Peng Chen, E-mail: pengchen729@sina.com
About author:Longjiang Gu, E-mail: gulongjiang0@163.com; Yan Wu, E-mail: yanwu9977@163.com
Supported by:
Chen's research was supported by the NSFC (11301297), the Hubei Provincial Natural Science Foundation of China (2024AFB730), the Yichang City Natural Science Foundation (A-24-3-008) and the Open Research Fund of Key Laboratory of Nonlinear Analysis and Applications (Central China Normal University), Ministry of Education, P. R. China (NAA2024ORG003). Gu's research was supported by the Zhejiang Provincial Natural Science Foundation (LQ21A010014) and the NFSC (12101577).

摘要： In this paper, we study the elliptic system
$\begin{equation*} \left\{ \begin{array}{ll} -\Delta u +V(x) u=|v|^{p-2}v-\lambda_2|v|^{s_2-2}v, \\ -\Delta v + V(x)v=|u|^{p-2}u-\lambda_1|u|^{s_1-2}u, \\ u,v \in H^1(\mathbb{R}^N) \end{array} \right.\end{equation*}$
with strongly indefinite structure and sign-changing nonlinearity. We overcome the absence of the upper semi-continuity assumption which is crucial in traditional variational methods for strongly indefinite problems. By some new tools and techniques we proved the existence of infinitely many geometrically distinct solutions if parameters $\lambda_1, \lambda_2>0$ small enough. To the best of our knowledge, our result seems to be the first result about infinitely many solutions for Hamiltonian system involving sign-changing nonlinearity.

关键词: variational method, strongly indefinite, elliptic system, multiple solutions, sign-changing nonlinearity

Abstract: In this paper, we study the elliptic system
$\begin{equation*} \left\{ \begin{array}{ll} -\Delta u +V(x) u=|v|^{p-2}v-\lambda_2|v|^{s_2-2}v, \\ -\Delta v + V(x)v=|u|^{p-2}u-\lambda_1|u|^{s_1-2}u, \\ u,v \in H^1(\mathbb{R}^N) \end{array} \right.\end{equation*}$
with strongly indefinite structure and sign-changing nonlinearity. We overcome the absence of the upper semi-continuity assumption which is crucial in traditional variational methods for strongly indefinite problems. By some new tools and techniques we proved the existence of infinitely many geometrically distinct solutions if parameters $\lambda_1, \lambda_2>0$ small enough. To the best of our knowledge, our result seems to be the first result about infinitely many solutions for Hamiltonian system involving sign-changing nonlinearity.

Key words: variational method, strongly indefinite, elliptic system, multiple solutions, sign-changing nonlinearity

中图分类号:

35J20

Peng Chen, Longjiang Gu, Yan Wu. MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION[J]. 数学物理学报(英文版), 2025, 45(2): 602-614.

Peng Chen, Longjiang Gu, Yan Wu. MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION[J]. Acta mathematica scientia,Series B, 2025, 45(2): 602-614.

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