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

  • Peng CHEN ,
  • Longjiang GU ,
  • Yan WU
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  • 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
Longjiang Gu, E-mail: gulongjiang0@163.com; Yan Wu, E-mail: yanwu9977@163.com

Received date: 2024-02-28

  Online published: 2025-05-08

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).

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.

Cite this article

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 . DOI: 10.1007/s10473-025-0218-3

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