{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:25:46Z","timestamp":1760145946796,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,10]],"date-time":"2024-09-10T00:00:00Z","timestamp":1725926400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"\u201cNonlinear Differential System in Applied Sciences\u201d of the Romanian Ministry of Research, Innovation and Digitization","award":["PNRR-III-C9-2022-I8 (Grant No. 22)"],"award-info":[{"award-number":["PNRR-III-C9-2022-I8 (Grant No. 22)"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider a Dirichlet problem driven by the anisotropic (p,q) Laplacian. In the reaction, we have a parametric partially concave term plus a \u201csuperlinear\u201d perturbation (convex term) which need not satisfy the Ambrosetti\u2013Rabinowitz condition. Using variational tools, we show that for all small values of the parameter \u03bb&gt;0, the problem has at least two nontrivial smooth solutions.<\/jats:p>","DOI":"10.3390\/sym16091188","type":"journal-article","created":{"date-parts":[[2024,9,10]],"date-time":"2024-09-10T09:10:57Z","timestamp":1725959457000},"page":"1188","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Anisotropic (p, q) Equation with Partially Concave Terms"],"prefix":"10.3390","volume":"16","author":[{"given":"Leszek","family":"Gasi\u0144ski","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of the National Education Commission, Krakow, Podchorazych 2, 30-084 Krakow, Poland"}]},{"given":"Gregoris","family":"Makrides","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the National Education Commission, Krakow, Podchorazych 2, 30-084 Krakow, Poland"}]},{"given":"Nikolaos S.","family":"Papageorgiou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece"},{"name":"Department of Mathematics, University of Craiova, 200585 Craiova, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1006\/jfan.1994.1078","article-title":"Combined effects of concave and convex nonlinearities in some elliptic problems","volume":"122","author":"Ambrosetti","year":"1994","journal-title":"J. Funct. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1142\/S0219199700000190","article-title":"Sobolev versus H\u00f6lder local minimizers and global multiplicity for some quasilinear elliptic equations","volume":"2","author":"Manfredi","year":"2000","journal-title":"Commun. Contemp. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/S0022-247X(03)00282-8","article-title":"W1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations","volume":"286","author":"Guo","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111861","DOI":"10.1016\/j.na.2020.111861","article-title":"Anisotropic equations with indefinite potential and competing nonlinearities","volume":"201","author":"Papageorgiou","year":"2020","journal-title":"Nonlinear Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1007\/s13163-022-00432-3","article-title":"Anisotropic Dirichlet double phase problems with competing nonlinearities","volume":"36","author":"Leonardi","year":"2023","journal-title":"Rev. Mat. Complut."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Diening, L., Harjulehto, P., H\u00e4st\u00f6, P., and R\u016f\u017ei\u010dka, M. (2011). Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer.","DOI":"10.1007\/978-3-642-18363-8"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1843","DOI":"10.1016\/S0362-546X(02)00150-5","article-title":"Existence of solutions for p(x)-Laplacian Dirichlet problem","volume":"52","author":"Fan","year":"2003","journal-title":"Nonlinear Anal."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Papageorgiou, N.S., and Winkert, P. (2018). Applied Nonlinear Functional Analysis, De Gruyter.","DOI":"10.1515\/9783110532982"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/j.jde.2007.01.008","article-title":"Global C1,\u03b1 regularity for variable exponent elliptic equations in divergence form","volume":"235","author":"Fan","year":"2007","journal-title":"J. Differ. Equ."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/9\/1188\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:53:07Z","timestamp":1760111587000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/9\/1188"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,10]]},"references-count":9,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["sym16091188"],"URL":"https:\/\/doi.org\/10.3390\/sym16091188","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,9,10]]}}}