{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:12:21Z","timestamp":1760235141077,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T00:00:00Z","timestamp":1627689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20180500"],"award-info":[{"award-number":["BK20180500"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2018YFC1508100"],"award-info":[{"award-number":["2018YFC1508100"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of China","award":["11701595"],"award-info":[{"award-number":["11701595"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schr\u00f6dinger\u2013Kirchhoff type equations a+b\u222b\u03a9\u00d7\u03a9|\u03be(x)\u2212\u03be(y)|p|x\u2212y|N+ps(x,y)dxdyp\u22121(\u2212\u0394)ps(\u00b7)\u03be+\u03bbV(x)|\u03be|p\u22122\u03be=f(x,\u03be),x\u2208\u03a9,\u03be=0,x\u2208\u2202\u03a9, where \u03a9 is a bounded Lipschitz domain in RN, 1&lt;p&lt;+\u221e, a,b&gt;0 are constants, s(\u00b7):RN\u00d7RN\u2192(0,1) is a continuous and symmetric function with N&gt;s(x,y)p for all (x,y)\u2208\u03a9\u00d7\u03a9, \u03bb&gt;0 is a parameter, (\u2212\u0394)ps(\u00b7) is a fractional p-Laplace operator with variable-order, V(x):\u03a9\u2192R+ is a potential function, and f(x,\u03be):\u03a9\u00d7RN\u2192R is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti\u2013Rabinowitz ((AR) for short) condition.<\/jats:p>","DOI":"10.3390\/sym13081393","type":"journal-article","created":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T21:46:44Z","timestamp":1627854404000},"page":"1393","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Infinitely Many Solutions for Fractional p-Laplacian Schr\u00f6dinger\u2013Kirchhoff Type Equations with Symmetric Variable-Order"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4655-3386","authenticated-orcid":false,"given":"Weichun","family":"Bu","sequence":"first","affiliation":[{"name":"College of Science, Hohai University, Nanjing 210098, China"},{"name":"College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tianqing","family":"An","sequence":"additional","affiliation":[{"name":"College of Science, Hohai University, Nanjing 210098, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6986-948X","authenticated-orcid":false,"given":"Jos\u00e9 Vanteler da C.","family":"Sousa","sequence":"additional","affiliation":[{"name":"Center for Mathematics, Computing and Cognition, Federal University of ABC, Avenida dos Estados, 5001, Bairro Bangu, Santo Andr\u00e9 09210-580, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3848-4456","authenticated-orcid":false,"given":"Yongzhen","family":"Yun","sequence":"additional","affiliation":[{"name":"College of Science, Hohai University, Nanjing 210098, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,31]]},"reference":[{"key":"ref_1","first-page":"101","article-title":"Existence results for fractional p-Laplacian problems via Morse theory","volume":"92","author":"Iannizzotto","year":"2014","journal-title":"Advan. 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