{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T11:23:15Z","timestamp":1771672995105,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,5]],"date-time":"2021-07-05T00:00:00Z","timestamp":1625443200000},"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"}]},{"name":"Innovation  Research  for  the  Postgraduates  of  Guangzhou  University","award":["2020GDJC-D07"],"award-info":[{"award-number":["2020GDJC-D07"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider a perturbed partial discrete Dirichlet problem with the (p,q)-Laplacian operator. Using critical point theory, we study the existence of infinitely many small solutions of boundary value problems. Without imposing the symmetry at the origin on the nonlinear term f, we obtain the sufficient conditions for the existence of infinitely many small solutions. As far as we know, this is the study of perturbed partial discrete boundary value problems. Finally, the results are exemplified by an example.<\/jats:p>","DOI":"10.3390\/sym13071207","type":"journal-article","created":{"date-parts":[[2021,7,5]],"date-time":"2021-07-05T22:02:04Z","timestamp":1625522524000},"page":"1207","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with (p,q)-Laplacian Operator"],"prefix":"10.3390","volume":"13","author":[{"given":"Feng","family":"Xiong","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":[[2021,7,5]]},"reference":[{"key":"ref_1","unstructured":"Elaydi, S. 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