{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:24:08Z","timestamp":1760239448399,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,6]],"date-time":"2020-11-06T00:00:00Z","timestamp":1604620800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971126"],"award-info":[{"award-number":["11971126"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018621","name":"Program for Changjiang Scholars and Innovative Research Team in University","doi-asserted-by":"publisher","award":["IRT_16R16"],"award-info":[{"award-number":["IRT_16R16"]}],"id":[{"id":"10.13039\/501100018621","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, based on critical point theory, we mainly focus on the multiplicity of nontrivial solutions for a nonlinear discrete Dirichlet boundary value problem involving the mean curvature operator. Without imposing the symmetry or oscillating behavior at infinity on the nonlinear term f, we respectively obtain the sufficient conditions for the existence of at least three non-trivial solutions and the existence of at least two non-trivial solutions under different assumptions on f. In addition, by using the maximum principle, we also deduce the existence of at least three positive solutions from our conclusion. As far as we know, our results are supplements to some well-known ones.<\/jats:p>","DOI":"10.3390\/sym12111839","type":"journal-article","created":{"date-parts":[[2020,11,8]],"date-time":"2020-11-08T20:23:35Z","timestamp":1604867015000},"page":"1839","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with \u03d5c-Laplacian"],"prefix":"10.3390","volume":"12","author":[{"given":"Yanshan","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China"},{"name":"Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5114-1418","authenticated-orcid":false,"given":"Zhan","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China"},{"name":"Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,6]]},"reference":[{"key":"ref_1","unstructured":"Elaydi, S. (2005). An Introduction to Difference Equations, Springer Science & Business Media."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1549","DOI":"10.1080\/10236198.2019.1669578","article-title":"Modeling Wolbachia infection in mosquito population via discrete dynamical models","volume":"25","author":"Yu","year":"2019","journal-title":"J. Differ. Equ. Appl."},{"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","unstructured":"Agarwal, R.P. (1992). Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1515\/JAA.2005.35","article-title":"A generalized upper and lower solution method for singular discrete boundary value problems for the one-dimensional p-Laplacian","volume":"11","author":"Jiang","year":"2005","journal-title":"J. Appl. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"145","DOI":"10.21136\/MB.2006.134087","article-title":"Existence and multiplicity results for nonlinear second order difference equations with Dirichlet boundary conditions","volume":"131","author":"Bereanu","year":"2006","journal-title":"Math. Bohem."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1768","DOI":"10.1080\/10236198.2019.1694014","article-title":"Multiple solutions for nonlinear functional difference equations by the invariant sets of descending flow","volume":"25","author":"Long","year":"2019","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1007\/BF02884022","article-title":"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_9","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.jmaa.2012.09.047","article-title":"Periodic and Neumann problems for discrete p(\u00b7)-Laplacian","volume":"399","author":"Bereanu","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/j.jmaa.2009.07.026","article-title":"Existence of gap solitons in periodic discrete nonlinear Schr\u00f6dinger equations","volume":"361","author":"Shi","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.aml.2015.06.012","article-title":"Existence of breathers for discrete nonlinear Schr\u00f6dinger equations","volume":"50","author":"Shi","year":"2015","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.aml.2016.08.001","article-title":"Homoclinic solutions in non-periodic discrete \u03d5-Laplacian equations with mixed nonlinearities","volume":"64","author":"Lin","year":"2017","journal-title":"Appl. Math. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.1016\/j.jde.2010.03.010","article-title":"On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems","volume":"249","author":"Zhou","year":"2010","journal-title":"J. Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1007\/s11425-014-4883-2","article-title":"Multiplicity results of breathers for the discrete nonlinear Schr\u00f6dinger equations with unbounded potentials","volume":"58","author":"Zhou","year":"2015","journal-title":"Sci. China Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/10236198.2019.1572128","article-title":"A note on homoclinic solutions of (p,q)-Laplacian difference equations","volume":"25","author":"Nastasi","year":"2019","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2985","DOI":"10.1016\/j.cam.2009.11.044","article-title":"On variational methods for nonlinear difference equations","volume":"233","author":"Galewski","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_18","first-page":"1","article-title":"Three solutions to Dirichlet boundary value problems for p-Laplacian difference equations","volume":"2008","author":"Jiang","year":"2007","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.aml.2018.03.006","article-title":"Existence of multiple solutions to second-order discrete Neumann boundary value problems","volume":"83","author":"Long","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3635","DOI":"10.1016\/j.amc.2010.10.019","article-title":"A fourth-order boundary value problem for a Sturm\u2013Liouville type equation","volume":"217","author":"Bonanno","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_21","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. Mathe. Lett."},{"key":"ref_22","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_23","doi-asserted-by":"crossref","first-page":"449","DOI":"10.4171\/zaa\/1573","article-title":"Two non-zero solutions for elliptic Dirichlet problems","volume":"35","author":"Bonanno","year":"2016","journal-title":"Z. Anal. Anwend."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.na.2003.11.012","article-title":"Multiple positive solutions of singular and nonsingular discrete problems via variational methods","volume":"58","author":"Agarwal","year":"2004","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_25","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."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3180","DOI":"10.1016\/j.na.2008.04.021","article-title":"Nonlinear difference equations investigated via critical point methods","volume":"70","author":"Bonanno","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_27","first-page":"1","article-title":"Positive solutions of discrete boundary value problems with the (p,q)-Laplacian operator","volume":"2017","author":"Nastasi","year":"2017","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s00030-014-0289-7","article-title":"Existence results for parametric boundary value problems involving the mean curvature operator","volume":"22","author":"Bonanno","year":"2015","journal-title":"Nonlinear Differ. Equ. Appl. NoDEA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.jde.2007.05.031","article-title":"Classical and non-classical solutions of a prescribed curvature equation","volume":"243","author":"Bonheure","year":"2007","journal-title":"J. Differ. Equ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.jmaa.2013.04.003","article-title":"Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space","volume":"405","author":"Corsato","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_31","first-page":"1","article-title":"Global bifurcation for problem with mean curvature operator on general domain","volume":"3","author":"Dai","year":"2017","journal-title":"Nonlinear Differ. Equ. Appl. NoDEA"},{"key":"ref_32","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_33","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."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1839\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:30:07Z","timestamp":1760178607000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1839"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,6]]},"references-count":33,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["sym12111839"],"URL":"https:\/\/doi.org\/10.3390\/sym12111839","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,11,6]]}}}