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

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

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.

Keywords

biharmonic-Kirchhoff system / symmetric mountain pass theorem / locally finite graph / infinitely many solutions / biharmonic system

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] A. Grigor’yan, Y. Lin, Y. Yang, Yamabe type equations on graphs, J. Differ. Equ., 261(2016), 4924-4943. https://doi.org/10.1016/j.jde.2016.07.011

[2] X. Han, M. Shao, L. Zhao, Existence and convergence of solutions for nonlinear biharmonic equations on graphs, J. Differ. Equ., 268(2020), 3936-3961. https://doi.org/10.1016/j.jde.2019.10.007

[3] A. Elmoataz, X. Desquesnes, O. Lezoray, Non-local morphological PDEs and p-Laplacian equation on graphs with applications in image processing and machine learning, IEEE J. Sel. Top. Signal Process., 6(2012), 764-779. https://doi.org/10.1109/JSTSP.2012.2216504

[4] A. Elmoataz, X. Desquesnes, M. Toutain, On the game p-Laplacian on weighted graphs with applications in image processing and data clustering, Eur. J. Appl. Math., 28(2017), 922-948. https://doi.org/10.1017/S0956792517000122

[5] A. Elmoataz, F. Lozes, M. Toutain, Nonlocal PDEs on graphs: From tug-of-war games to unified interpolation on images and point clouds, J. Math. Imaging Vision, 57(2017), 381-401. https://doi.org/10.1007/s10851-016-0683-3

[6] H. Ennaji, Y. Quéau, A. Elmoataz, Tug of war games and PDEs on graphs with applications in image and high dimensional data processing, Sci. Rep., 13(2023), 6045. https://doi.org/10.1038/s41598-023-32354-5

[7] A. Elmoataz, M. Toutain, D. Tenbrinck, On the p-Laplacian and ∞-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci., 8(2015), 2412-2451. https://doi.org/10.1137/15M1022793

[8] X. Han, M. Shao, p-Laplacian equations on locally finite graphs, Acta Math. Sin., 37(2021), 1645-1678. https://doi.org/10.1007/s10114-021-9523-5

[9] Y. Pang, J. Xie, X. Zhang, Infinitely many solutions for three quasilinear Laplacian systems on weighted graphs, Boundary Value Probl., 2024 (2024), 45. https://doi.org/10.1186/s13661-024-01846-2

[10] M. Shao, Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs, J. Fixed Point Theory Appl., 25(2023), 50. https://doi.org/10.1007/s11784-023-01055-x

[11] M. Shao, Existence and multiplicity of solutions to p-Laplacian equations on graphs, Rev. Mat. Complutense, 37(2024), 185-203. https://doi.org/10.1007/s13163-022-00452-z

[12] P. Yang, X. Zhang, Existence and multiplicity of nontrivial solutions for a (p, q)- Laplacian system on locally finite graphs, Taiwanese J. Math., 28(2024), 551-588. https://doi.org/10.11650/tjm/240201

[13] X. Zhang, X. Zhang, Existence and convergence of ground state solutions for a (p, q)-Laplacian system on weighted graphs, Asymptotic Anal., 141(2025), 178-208. https://doi.org/10.1177/09217134241308436

[14] X. Ou, X. Zhang, Least energy sign-changing solutions for Kirchhoff-type equations with logarithmic nonlinearity on locally finite graphs, TWMS J. Pure Appl. Math., 15(2024), 286-317. https://doi.org/10.30546/2219-1259.15.2.2024.01286

[15] G. Pan, C. Ji, Existence and convergence of the least energy sign-changing solutions for nonlinear Kirchhoff equations on locally finite graphs, Asymptot. Anal., 133(2023), 463-482. https://doi.org/10.3233/ASY-221819

[16] X. Wang, J. Xie, X. Zhang, X. Ou, Dependence on parameters of solutions and optimal control for a generalized poly-Laplacian system on weighted graphs, Electron. Res. Arch., 34(2026), 5303-5327. https://doi.org/10.3934/era.2026236

[17] Z. Yu, J. Xie, X. Zhang, Existence and multiplicity of solutions for a class of (p, q)-Kirchhoff system with combined nonlinearities on graphs, Boundary Value Probl., 2024 (2024), 134. https://doi.org/10.1186/s13661-024-01947-y

[18] W. Bu, T. An, Y. Li, J. He, Kirchhoff-type problems involving logarithmic nonlinearity with variable exponent and convection term, Mediterr. J. Math., 20(2023), 77. https://doi.org/10.1007/s00009-023-02273-w

[19] Y. Gao, Y. Jiang, L. S. Liu, N. Wei, Multiple positive solutions for a logarithmic Kirchhoff type problem in R³ , Appl. Math. Lett., 139(2023), 108539. https://doi.org/10.1016/j.aml.2022.108539

[20] Y. Li, D. Wang, J. Zhang, Sign-changing solutions for a class of p-Laplacian Kirchhoff- type problem with logarithmic nonlinearity, AIMS Math., 5(2020), 2100-2112. https://doi.org/10.3934/math.2020139

[21] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differ. Equations, 259(2015), 1256-1274. https://doi.org/10.1016/j.jde.2015.02.040

[22] L. Wen, X. Tang, S. Chen, Ground state sign-changing solutions for Kirchhoff equations with logarithmic nonlinearity, Electron. J. Qual. Theory Differ. Equ., 47(2019), 1-13. https://doi.org/10.14232/ejqtde.2019.1.47

[23] J. Kuang, S. Cai, Periodic solutions with prescribed minimal period for distributed delay differential equations, Discrete Contin. Dyn. Syst. Ser. S, 21(2026), 148-170. https://doi.org/10.3934/dcdss.2026020

[24] J. Liu, S. Chen, X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in R^N , J. Math. Anal. Appl., 395(2012), 608-615. https://doi.org/10.1016/j.jmaa.2012.05.063

[25] Y. Wei, Multiplicity results for some fourth-order elliptic equations, J. Math. Anal. Appl., 385(2012), 797-807. https://doi.org/10.1016/j.jmaa.2011.07.011

[26] Y. Ye, C. Tang, Infinitely many solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 394(2012), 841-854. https://doi.org/10.1016/j.jmaa.2012.04.041

[27] Z. Yuan, S. Liu, Existence and multiplicity of solutions for a biharmonic Kirchhoff equation in R⁵ , J. Nonlinear Model. Anal., 6(2024), 71-87. https://doi.org/10.12150/jnma.2024.71

[28] J. Zhang, Z. Wei, Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems, Nonlinear Anal., 74(2011), 7474-7485. https://doi.org/10.1016/j.na.2011.07.067

[29] W. Zhang, X. H. Tang, J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential, Taiwanese J. Math., 18(2014), 645-659. https://doi.org/10.11650/tjm.18.2014.3584

[30] L. Zhao, N. Zhang, Existence of solutions for a higher order Kirchhoff type problem with exponential critical growth, Nonlinear Anal., 132(2016), 214-226. https://doi.org/10.1016/j.na.2015.11.014

[31] C. Ji, V. D. R˘adulescu, Ground states of biharmonic equations on lattice graphs, Topol. Methods Nonlinear Anal., 66(2025), 273-288. https://doi.org/10.12775/TMNA.2025.014

[32] C. Ji, O. H. Miyagaki, Existence of ground state solutions for the nonlinear elliptic equations with zero mass on lattice graphs, Topol. Methods Nonlinear Anal., 66(2025), 109-127. https://doi.org/10.12775/TMNA.2025.005

[33] Z. He, C. Ji, The double phase problems on lattice graphs, J. Differ. Equ., 438(2025), 113380. https://doi.org/10.1016/j.jde.2025.113380

[34] Z. He, C. Ji, Existence and multiplicity of solutions for the logarithmic Schrödinger equation with a potential on lattice graphs, J. Geom. Anal., 34(2024), 378. https://doi.org/10.1007/s12220-024-01829-w

[35] Y. Ding, C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differ. Equ., 222(2006), 137-163. https://doi.org/10.1016/j.jde.2005.03.011

[36] X. H. Tang, Infinitely many solutions for semilinear Schrödinger equations with sign- changing potential and nonlinearity, J. Math. Anal. Appl., 401(2013), 407-415. https://doi.org/10.1016/j.jmaa.2012.12.035

[37] G. Bonanno, G. M. Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Boundary Value Probl., 2009 (2009), 670675. https://doi.org/10.1155/2009/670675

[38] M. Shao, Partial Differential Equations on Graphs, Ph.D thesis, Tsinghua University, 2020.

[39] G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitate, Rend. Instituto Lombardo Sci. Lett., 112(1978), 332-336.

[40] T. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal., 7(1983), 981-1012. https://doi.org/10.1016/0362-546X(83)90115-3

[41] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, RI, 1986. https://doi.org/10.1090/cbms/065