{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:41:20Z","timestamp":1753886480900,"version":"3.41.2"},"reference-count":16,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,5,2]],"date-time":"2011-05-02T00:00:00Z","timestamp":1304294400000},"content-version":"vor","delay-in-days":121,"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>This paper considers bifurcation at the principal eigenvalue of a class of gradient operators which possess the Palais\u2010Smale condition. The existence of the bifurcation branch and the asymptotic nature of the bifurcation is verified by using the compactness in the Palais Smale condition and the order of the nonlinearity in the operator. The main result is applied to estimate the asyptotic behaviour of solutions to a class of semilinear elliptic equations with a critical Sobolev exponent.<\/jats:p>","DOI":"10.1155\/2011\/564930","type":"journal-article","created":{"date-parts":[[2011,5,2]],"date-time":"2011-05-02T19:31:47Z","timestamp":1304364707000},"update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bifurcation of Gradient Mappings Possessing the Palais\u2010Smale Condition"],"prefix":"10.1155","volume":"2011","author":[{"given":"Elliot","family":"Tonkes","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2011,5,2]]},"reference":[{"volume-title":"Topological Methods in the Theory of Nonlinear Integral Equations","year":"1964","author":"Krasnosel\u2032skii M. A.","key":"e_1_2_5_1_2"},{"key":"e_1_2_5_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02412022"},{"key":"e_1_2_5_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01112603"},{"key":"e_1_2_5_4_2","article-title":"La biforcazione nel caso variazionale","author":"Marino A.","year":"1977","journal-title":"Conferenze del Seminario di Matematica dell\u2032Universit\u00e0 di Bari"},{"key":"e_1_2_5_5_2","doi-asserted-by":"publisher","DOI":"10.1090\/cbms\/065"},{"key":"e_1_2_5_6_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0017089500032080"},{"key":"e_1_2_5_7_2","series-title":"Oxford Mathematical Monographs","volume-title":"Spectral Theory and Differential Operators","author":"Edmunds D. E.","year":"1987"},{"key":"e_1_2_5_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2008.12.061"},{"key":"e_1_2_5_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2010.05.050"},{"key":"e_1_2_5_10_2","doi-asserted-by":"publisher","DOI":"10.1017\/S001708950400206X"},{"key":"e_1_2_5_11_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0017089508004588"},{"key":"e_1_2_5_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2061-7"},{"key":"e_1_2_5_13_2","article-title":"On the conditions for extremals of functionals","volume":"41","author":"Lusternik L. A.","year":"1934","journal-title":"Matematicheskii Sbornik"},{"key":"e_1_2_5_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8159-4"},{"key":"e_1_2_5_15_2","doi-asserted-by":"publisher","DOI":"10.57262\/die\/1369055606"},{"key":"e_1_2_5_16_2","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160360405"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/downloads.hindawi.com\/journals\/ijmms\/2011\/564930.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/downloads.hindawi.com\/journals\/ijmms\/2011\/564930.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2011\/564930","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,10]],"date-time":"2024-06-10T13:46:23Z","timestamp":1718027183000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/onlinelibrary.wiley.com\/doi\/10.1155\/2011\/564930"}},"subtitle":[],"editor":[{"given":"Marco","family":"Squassina","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1155\/2011\/564930"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1155\/2011\/564930","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2011,1]]},"assertion":[{"value":"2010-12-30","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-03-04","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-05-02","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"564930"}}