{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:34:12Z","timestamp":1760150052305,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,25]],"date-time":"2023-09-25T00:00:00Z","timestamp":1695600000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NNSF of China","doi-asserted-by":"publisher","award":["12061016"],"award-info":[{"award-number":["12061016"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, by using critical point theory, the existence of infinitely many small solutions for a perturbed partial discrete Dirichlet problems including the mean curvature operator is investigated. Moreover, the present study first attempts to address discrete Dirichlet problems with \u03d5c-Laplacian operator in relative to some relative existing references. Based on our knowledge, this is the research of perturbed partial discrete bvp with \u03d5c-Laplacian operator for the first time. At last, two examples are used to examplify the results.<\/jats:p>","DOI":"10.3390\/axioms12100909","type":"journal-article","created":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T03:45:23Z","timestamp":1695699923000},"page":"909","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Infinitely Many Solutions for a Perturbed Partial Discrete Dirichlet Problem Involving \u03d5c-Laplacian"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-4288-7414","authenticated-orcid":false,"given":"Feng","family":"Xiong","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"},{"name":"College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4672","DOI":"10.1016\/j.na.2011.11.018","article-title":"Periodic solutions of second order nonlinear difference systems with \u03d5-Laplacian: A variational approach","volume":"75","author":"Mawhin","year":"2012","journal-title":"Nonlinear Anal."},{"key":"ref_2","unstructured":"Elaydi, S. (2005). An Introduction to Difference Equations, Springer. [3rd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105117","DOI":"10.1016\/j.cnsns.2019.105117","article-title":"Global dynamics of a delayed two-patch discrete SIR disease model","volume":"83","author":"Long","year":"2020","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1549","DOI":"10.1080\/10236198.2019.1669578","article-title":"Modeling Wolbachia infection in mosquito population via discrete dynamical model","volume":"25","author":"Yu","year":"2019","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_5","unstructured":"Agarwal, R. (1992). Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1007\/BF02884022","article-title":"The existence of periodic and subharmonic solutions for second-order superlinear difference equations","volume":"46","author":"Guo","year":"2003","journal-title":"Sci. China Ser. A Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s12190-014-0796-z","article-title":"Periodic and subharmonic solutions for second-order nonlinear difference equations","volume":"48","author":"Shi","year":"2015","journal-title":"J. Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s11425-010-4101-9","article-title":"Homoclinic solutions in periodic difference equations with saturable nonlinearity","volume":"54","author":"Zhou","year":"2011","journal-title":"Sci. China Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s10884-019-09743-4","article-title":"Ground state solutions of discrete asymptotically linear Schr\u00f6dinger equations with bounded and non-periodic potentials","volume":"32","author":"Lin","year":"2020","journal-title":"J. Dynam. Differ. Equ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.aml.2014.10.006","article-title":"Boundary value problems for 2n-order \u03d5c-Laplacian difference equations containing both advance and retardation","volume":"41","author":"Zhou","year":"2015","journal-title":"Appl. Math. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1080\/00036810902942242","article-title":"Infinitely many solutions for a class of discrete non-linear boundary value problems","volume":"88","author":"Bonanno","year":"2009","journal-title":"Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.aml.2015.09.005","article-title":"Superlinear discrete problems","volume":"52","author":"Bonanno","year":"2016","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.jmaa.2016.10.023","article-title":"Positive solutions for a discrete two point nonlinear boundary value problem with p-Laplacian","volume":"447","author":"Mawhin","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","first-page":"1","article-title":"Positive solutions of discrete boundary value problems with the (p,q)-Laplacian operator","volume":"225","author":"Nastasi","year":"2017","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.aml.2018.11.016","article-title":"Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with \u03d5c-Laplacian","volume":"91","author":"Zhou","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_16","first-page":"3183","article-title":"Positive solutions of the discrete Robin problem with \u03d5-Laplacian","volume":"14","author":"Ling","year":"2021","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1515\/math-2019-0081","article-title":"Positive solutions of the discrete Dirichlet problem involving the mean curvature operator","volume":"17","author":"Ling","year":"2019","journal-title":"Open Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wang, J., and Zhou, Z. (2020). Large constant-sign solutions of discrete Dirichlet boundary value problems with p-mean curvature operator. Mathematics, 8.","DOI":"10.3390\/math8030381"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1080\/10236198.2014.988619","article-title":"Multiple solutions for partial discrete Dirichlet problems depending on a real parameter","volume":"21","author":"Heidarkhani","year":"2015","journal-title":"J. Difference Equ. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Du, S., and Zhou, Z. (2020). Multiple solutions for partial discrete Dirichlet problems involving the p-Laplacian. Mathematics, 8.","DOI":"10.3390\/math8112030"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1186\/s13661-021-01514-9","article-title":"Three solutions for a partial discrete Dirichlet boundary value problem with p-Laplacian","volume":"2021","author":"Wang","year":"2021","journal-title":"Bound. Value Probl."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Xiong, F., and Zhou, Z. (2021). Small solutions of the perturbed nonlinear partial discrete Dirichlet boundary value problems with (p,q)-Laplacian operator. Symmetry, 13.","DOI":"10.3390\/sym13071207"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1515\/anona-2020-0195","article-title":"On the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curature operator","volume":"11","author":"Du","year":"2022","journal-title":"Adv. Nonlinear Anal."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Xiong, F. (2023). Infinitely many solutions for partial discrete Kirchhoff type problems involving p-Laplacian. Mathematics, 11.","DOI":"10.3390\/math11153288"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Zhang, H., and Long, Y. (2023). Multiple Existence results of nontrivial solutions for a class of second-order partial difference equations. Symmetry, 15.","DOI":"10.3390\/sym15010006"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1515\/anona-2022-0251","article-title":"Nontrivial solutions of discrete Kirchhoff-type problems via Morse theory","volume":"11","author":"Long","year":"2022","journal-title":"Adv. Nonlinear Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1596","DOI":"10.3934\/era.2023082","article-title":"Multiple nontrivial periodic solutions to a second-order partial difference equation","volume":"31","author":"Long","year":"2023","journal-title":"Electron. Res. Arch."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1006\/jdeq.1996.0013","article-title":"On a modified capillary equation","volume":"124","author":"Clement","year":"1996","journal-title":"J. Differ. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1090\/S0002-9939-08-09612-3","article-title":"Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces","volume":"137","author":"Bereanu","year":"2009","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1515\/ans-2014-0406","article-title":"Variational methods on finite dimensional Banach spaces and discrete problems","volume":"14","author":"Bonanno","year":"2014","journal-title":"Adv. Nonlinear Stud."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1023\/B:JOGO.0000026447.51988.f6","article-title":"A critical points theorem and nonlinear differential problems","volume":"28","author":"Bonanno","year":"2004","journal-title":"J. Glob. Optim."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/909\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:57:33Z","timestamp":1760129853000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/909"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,25]]},"references-count":31,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["axioms12100909"],"URL":"https:\/\/doi.org\/10.3390\/axioms12100909","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,9,25]]}}}