Dependence on parameters of solutions and optimal control for a generalized poly-Laplacian system on weighted graphs

Xiaoyu Wang; Junping Xie; Xingyong Zhang; Xin Ou; Xiaoyu Wang; Junping Xie; Xingyong Zhang; Xin Ou

doi:10.3934/era.2026236

Electronic Research Archive

2026, Volume 34, Issue 8: 5303-5327. doi: 10.3934/era.2026236

Previous Article Next Article

Research article Special Issues

Dependence on parameters of solutions and optimal control for a generalized poly-Laplacian system on weighted 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: 23 February 2026 Revised: 27 May 2026 Accepted: 02 June 2026 Published: 22 June 2026

We mainly investigated the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph $ G = (V, E) $. We first presented an existence result of mountain pass type nontrivial solutions when the nonlinear term $ F $ satisfies the super-$ (p, q) $ linear growth condition, which is a generalization of those results in ^[6]. Then, we mainly showed that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters. Similarly, we also presented the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the local minimum type nontrivial solutions. Subsequently, we presented an example on optimal control as an application of our results. Finally, we gave a nonexistence result and some results for the corresponding scalar equation.
- generalized ploy-Laplacian system,
- dependence on parameters,
- weighted finite graph,
- mountain pass type solution,
- local minimum type solution,
- optimal control
Citation: Xiaoyu Wang, Junping Xie, Xingyong Zhang, Xin Ou. Dependence on parameters of solutions and optimal control for a generalized poly-Laplacian system on weighted graphs[J]. Electronic Research Archive, 2026, 34(8): 5303-5327. doi: 10.3934/era.2026236

Related Papers:

Abstract

We mainly investigated the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph $ G = (V, E) $. We first presented an existence result of mountain pass type nontrivial solutions when the nonlinear term $ F $ satisfies the super-$ (p, q) $ linear growth condition, which is a generalization of those results in ^[6]. Then, we mainly showed that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters. Similarly, we also presented the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the local minimum type nontrivial solutions. Subsequently, we presented an example on optimal control as an application of our results. Finally, we gave a nonexistence result and some results for the corresponding scalar equation.

References

[1]	H. Ge, W. Jiang, Yamabe equations on infinite graphs, J. Math. Anal. Appl., 460 (2018), 885–890. https://doi.org/10.1016/j.jmaa.2017.12.020 doi: 10.1016/j.jmaa.2017.12.020
[2]	A. Grigor'yan, Y. Lin, Y. Yang, Yamabe type equations on graphs, J. Differ. Equations, 261 (2016), 4924–4943. https://doi.org/10.1016/j.jde.2016.07.011 doi: 10.1016/j.jde.2016.07.011
[3]	H. Ge, A $p$-th Yamabe equation on graph, Proc. Am. Math. Soc., 146 (2018), 2219–2224. https://doi.org/10.1090/proc/13929 doi: 10.1090/proc/13929
[4]	X. Zhang, A. Lin, Positive solutions of $p$-th Yamabe type equations on graphs, Front. Math. China, 13 (2018), 1501–1514. https://doi.org/10.1007/s11464-018-0734-8 doi: 10.1007/s11464-018-0734-8
[5]	X. Zhang, A. Lin, Positive solutions of $p$-th Yamabe type equations on infinite graphs, Proc. Am. Math. Soc., 147 (2019), 1421–1427. https://doi.org/10.1090/proc/14362 doi: 10.1090/proc/14362
[6]	X. Zhang, X. Zhang, J. Xie, X. Yu, Existence and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs, Boundary Value Probl., 2022 (2022), 32. https://doi.org/10.1186/s13661-022-01613-1 doi: 10.1186/s13661-022-01613-1
[7]	X. Yu, X. Zhang, J. Xie, X. Zhang, Existence of nontrivial solutions for a class of poly-Laplacian system with mixed nonlinearity on graphs, Math. Methods Appl. Sci., 47 (2024), 1750–1763. https://doi.org/10.1002/mma.9621 doi: 10.1002/mma.9621
[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 doi: 10.1007/s10114-021-9523-5
[9]	A. El Chakik, A. Elmoataz, X. Desquesnes, Mean curvature flow on graphs for image and manifold restoration and enhancement, Signal Process., 105 (2014), 449–463. https://doi.org/10.1016/j.sigpro.2014.04.029 doi: 10.1016/j.sigpro.2014.04.029
[10]	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 doi: 10.1109/JSTSP.2012.2216504
[11]	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 doi: 10.1017/S0956792517000122
[12]	H. Ennaji, Y. Qu$\acute{e}$au, 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 doi: 10.1038/s41598-023-32354-5
[13]	A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. https://doi.org/10.1016/0022-1236(73)90051-7 doi: 10.1016/0022-1236(73)90051-7
[14]	K. Chang, Critical Point Theory and Its Applications (in Chinese), Shanghai Scientific & Technical Publishers, Shanghai, 1986.
[15]	P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, 1986. https://doi.org/10.1090/cbms/065
[16]	P. Yang, X. Zhang, Existence and multiplicity of nontrivial solutions for a $(p, q)$-Laplacian system on locally finite graphs, Taiwan. J. Math., 28 (2024), 551–588. https://doi.org/10.11650/tjm/240201 doi: 10.11650/tjm/240201
[17]	Y. Pang, X. Zhang, Existence of three solutions for two quasilinear Laplacian systems on graphs Demonstr. Math., 57 (2024), 20240062. https://doi.org/10.1515/dema-2024-0062. doi: 10.1515/dema-2024-0062
[18]	G. Bonanno, G. Bisci, V. Rǎdulescu, Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket, ESAIM. Control. Optim. Calc. Var., 18 (2012), 941–953. https://doi.org/10.1051/cocv/2011199 doi: 10.1051/cocv/2011199
[19]	B. Breckner, V. Rǎdulescu, C. Varga, Infinitely many solutions for the Dirichlet problem on the Sierpiński gasket, Anal. Appl., 9 (2011), 235–248. https://doi.org/10.1142/S0219530511001844 doi: 10.1142/S0219530511001844
[20]	K. Falconer, J. Hu, Nonlinear elliptical equations on the Sierpiński gasket, J. Math. Anal. Appl., 240 (1999), 552–573. https://doi.org/10.1006/jmaa.1999.6617 doi: 10.1006/jmaa.1999.6617
[21]	M. Galewski, On the mountain pass solution to boundary value problems on the Sierpiński gasket, Results Math., 74 (2019), 167. https://doi.org/10.1007/s00025-019-1092-x doi: 10.1007/s00025-019-1092-x
[22]	M. Galewski, J. Smejda, On the dependence on parameters for mountain pass solutions of second order discrete BVP's, Appl. Math. Comput., 219 (2013), 5963–5971. https://doi.org/10.1016/j.amc.2012.12.028 doi: 10.1016/j.amc.2012.12.028
[23]	M. Galewski, Dependence on parameters for a discrete Emden-Fowler equation, Appl. Math. Comput., 218 (2011), 1247–1253. https://doi.org/10.1016/j.amc.2011.06.005 doi: 10.1016/j.amc.2011.06.005
[24]	M. Galewski, Dependence on parameters for discrete second-order boundary value problems, J. Differ. Equations Appl., 17 (2011), 1441–1453. https://doi.org/10.1080/10236191003639442 doi: 10.1080/10236191003639442
[25]	J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4757-2061-7
[26]	X. Zhang, X. Tang, Non-constant periodic solutions for second order Hamiltonian system with a p-Laplacian, Math. Slovaca, 62 (2012), 231–246. https://doi.org/10.2478/s12175-012-0005-1 doi: 10.2478/s12175-012-0005-1
[27]	P. Lindqvist, On the equation $\mbox{div}(\|\nabla u\|^{p-2}\nabla u)+\lambda \|u\|^{p-2}u = 0$, Proc. Am. Math. Soc., 109 (1990), 157–164. https://doi.org/10.2307/2048375 doi: 10.2307/2048375

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)