{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:15:20Z","timestamp":1760145320256,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T00:00:00Z","timestamp":1719964800000},"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":["11201213","ZR2015AM026"],"award-info":[{"award-number":["11201213","ZR2015AM026"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"NSF of Shandong Province","award":["11201213","ZR2015AM026"],"award-info":[{"award-number":["11201213","ZR2015AM026"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A class of semi-linear elliptic equations with the critical Hardy\u2013Sobolev exponent has been considered. This model is widely used in hydrodynamics and glaciology, gas combustion in thermodynamics, quantum field theory, and statistical mechanics, as well as in gravity balance problems in galaxies. The PSc sequence of energy functional was investigated, and then the mountain pass lemma was used to prove the existence of at least one nontrivial solution. Also a multiplicity result was obtained. Some known results were generalized.<\/jats:p>","DOI":"10.3390\/axioms13070450","type":"journal-article","created":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T09:23:59Z","timestamp":1719998639000},"page":"450","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Existence and Multiplicity of Nontrivial Solutions for Semilinear Elliptic Equations Involving Hardy\u2013Sobolev Critical Exponents"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9763-1575","authenticated-orcid":false,"given":"Yonghong","family":"Fan","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics Sciences, Ludong University, Yantai 264025, China"}]},{"given":"Wenheng","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Sciences, Ludong University, Yantai 264025, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6610-4406","authenticated-orcid":false,"given":"Linlin","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Sciences, Ludong University, Yantai 264025, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,3]]},"reference":[{"key":"ref_1","unstructured":"Dautray, R., and Lions, J.L. 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