Infinitely many solutions of biharmonic-Kirchhoff systems on locally finite graphs

Xiaoyu Wang; Junping Xie; Xingyong Zhang; Xiaoyu Wang; Junping Xie; Xingyong Zhang

doi:10.3934/nhm.2026069

Networks and Heterogeneous Media

2026, Volume 21, Issue 5: 1866-1889. doi: 10.3934/nhm.2026069

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Research article

Infinitely many solutions of biharmonic-Kirchhoff systems on locally finite graphs

1.
Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China
2.
Faculty of Transportation Engineering, Kunming University of Science and Technology, Kunming 650500, China
3.
Research Center for Mathematics and Interdisciplinary Sciences, Kunming University of Science and Technology, Kunming 650500, China

Received: 11 December 2025 Revised: 27 July 2026 Accepted: 18 August 2026 Published: 08 September 2026

In recent years, the study of partial differential equations (PDEs) and their systems in graph settings has emerged as a prominent research area, driven by their significant applications in image processing and data clustering. Motivated by these developments, we aim to extend certain existence results for biharmonic-Kirchhoff systems and biharmonic systems from the Euclidean setting (continuous models) to their counterparts in locally finite graphs (discrete models). Our primary focus is on establishing the existence of infinitely many solutions for a biharmonic-Kirchhoff system defined on a locally finite graph by employing the symmetric mountain pass theorem as our key analytical tool. Under the assumption that the nonlinear term is super-quartic, we prove that the system admits infinitely many solutions. Additionally, we present analogous results for the biharmonic system. A notable observation from our study is that the results obtained in the locally finite graph setting surpass those in the Euclidean setting. This improvement is fundamentally attributed to the more favorable embedding properties inherent to the locally finite graph framework.
- biharmonic-Kirchhoff system,
- symmetric mountain pass theorem,
- locally finite graph,
- infinitely many solutions,
- biharmonic system
Citation: Xiaoyu Wang, Junping Xie, Xingyong Zhang. Infinitely many solutions of biharmonic-Kirchhoff systems on locally finite graphs[J]. Networks and Heterogeneous Media, 2026, 21(5): 1866-1889. doi: 10.3934/nhm.2026069

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Abstract

In recent years, the study of partial differential equations (PDEs) and their systems in graph settings has emerged as a prominent research area, driven by their significant applications in image processing and data clustering. Motivated by these developments, we aim to extend certain existence results for biharmonic-Kirchhoff systems and biharmonic systems from the Euclidean setting (continuous models) to their counterparts in locally finite graphs (discrete models). Our primary focus is on establishing the existence of infinitely many solutions for a biharmonic-Kirchhoff system defined on a locally finite graph by employing the symmetric mountain pass theorem as our key analytical tool. Under the assumption that the nonlinear term is super-quartic, we prove that the system admits infinitely many solutions. Additionally, we present analogous results for the biharmonic system. A notable observation from our study is that the results obtained in the locally finite graph setting surpass those in the Euclidean setting. This improvement is fundamentally attributed to the more favorable embedding properties inherent to the locally finite graph framework.

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