{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T19:00:45Z","timestamp":1764961245784,"version":"3.46.0"},"reference-count":21,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T00:00:00Z","timestamp":1672531200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,8,24]]},"abstract":"Abstract<\/jats:title>\n \n We study general problems modeling electrostatic microelectromechanical systems devices\n \n \n (\n \n P\n \u03bb<\/jats:sub>\n <\/jats:italic>\n )\n <\/jats:label>\n \n \n \n \n \n \n \n \n \u03c6<\/m:mi>\n \n (<\/m:mo>\n \n r<\/m:mi>\n ,<\/m:mo>\n \u2212<\/m:mo>\n u<\/m:mi>\n \u2032<\/m:mo>\n \n (<\/m:mo>\n \n r<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n \u03bb<\/m:mi>\n \n \n \n \n \u222b<\/m:mo>\n <\/m:mstyle>\n <\/m:mrow>\n <\/m:mrow>\n \n 0<\/m:mn>\n <\/m:mrow>\n \n r<\/m:mi>\n <\/m:mrow>\n <\/m:munderover>\n \n \n f<\/m:mi>\n \n (<\/m:mo>\n \n s<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \n g<\/m:mi>\n \n (<\/m:mo>\n \n u<\/m:mi>\n \n (<\/m:mo>\n \n s<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mfrac>\n d<\/m:mi>\n s<\/m:mi>\n ,<\/m:mo>\n \n <\/m:mtd>\n \n r<\/m:mi>\n \u2208<\/m:mo>\n \n (<\/m:mo>\n \n 0<\/m:mn>\n ,<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n ,<\/m:mo>\n <\/m:mtd>\n <\/m:mtr>\n \n \n 0<\/m:mn>\n <<\/m:mo>\n u<\/m:mi>\n \n (<\/m:mo>\n \n r<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <<\/m:mo>\n 1<\/m:mn>\n ,<\/m:mo>\n \n <\/m:mtd>\n \n r<\/m:mi>\n \u2208<\/m:mo>\n \n (<\/m:mo>\n \n 0<\/m:mn>\n ,<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n ,<\/m:mo>\n <\/m:mtd>\n <\/m:mtr>\n \n \n u<\/m:mi>\n \n (<\/m:mo>\n \n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n 0<\/m:mn>\n ,<\/m:mo>\n \n <\/m:mtd>\n <\/m:mtr>\n <\/m:mtable>\n <\/m:mrow>\n <\/m:mfenced>\n <\/m:math>\n \\left\\{\\begin{array}{ll}\\varphi (r,-u^{\\prime} \\left(r))=\\lambda \\underset{0}{\\overset{r}{\\displaystyle \\int }}\\frac{f\\left(s)}{g\\left(u\\left(s))}{\\rm{d}}s,\\hspace{1.0em}& r\\in \\left(0,1),\\\\ 0\\lt u\\left(r)\\lt 1,\\hspace{1.0em}& r\\in \\left(0,1),\\\\ u\\left(1)=0,\\hspace{1.0em}\\end{array}\\right.<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n where\n \n \n \n \n \u03c6<\/m:mi>\n <\/m:math>\n \\varphi<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n \n \n \n \n g<\/m:mi>\n <\/m:math>\n g<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and\n \n \n \n \n f<\/m:mi>\n <\/m:math>\n f<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are some functions on\n \n \n \n \n \n [<\/m:mo>\n \n 0<\/m:mn>\n ,<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n ]<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \\left[0,1]<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n \n \n \u03bb<\/m:mi>\n ><\/m:mo>\n 0<\/m:mn>\n <\/m:math>\n \\lambda \\gt 0<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a parameter. We obtain results on the existence and regularity of a touchdown solution to (\n \n \n \n \n \n \n P<\/m:mi>\n <\/m:mrow>\n \n \u03bb<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n {P}_{\\lambda }<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when\n \n \n \n \n \u03c6<\/m:mi>\n \n (<\/m:mo>\n \n r<\/m:mi>\n ,<\/m:mo>\n v<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n \n \n r<\/m:mi>\n <\/m:mrow>\n \n \u03b1<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \n \n \u2223<\/m:mo>\n v<\/m:mi>\n \u2223<\/m:mo>\n <\/m:mrow>\n \n \u03b2<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n v<\/m:mi>\n <\/m:math>\n \\varphi \\left(r,v)={r}^{\\alpha }{| v| }^{\\beta }v<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , i.e., when the associated differential equation involves the operator\n \n \n \n \n \n \n r<\/m:mi>\n <\/m:mrow>\n \n \u2212<\/m:mo>\n \u03b3<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \n (<\/m:mo>\n \n \n \n r<\/m:mi>\n <\/m:mrow>\n \n \u03b1<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \n \n \u2223<\/m:mo>\n u<\/m:mi>\n \u2032<\/m:mo>\n \u2223<\/m:mo>\n <\/m:mrow>\n \n \u03b2<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n u<\/m:mi>\n \u2032<\/m:mo>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u2032<\/m:mo>\n <\/m:math>\n {r}^{-\\gamma }\\left({r}^{\\alpha }{| u^{\\prime} | }^{\\beta }u^{\\prime} )^{\\prime}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.\n <\/jats:p>","DOI":"10.1515\/anona-2023-0102","type":"journal-article","created":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T10:23:06Z","timestamp":1692872586000},"source":"Crossref","is-referenced-by-count":0,"title":["Touchdown solutions in general MEMS models"],"prefix":"10.1515","volume":"12","author":[{"given":"Rodrigo","family":"Clemente","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidade Federal Rural de Pernambuco , 52171-900 , Recife , Pernambuco , Brazil"}]},{"given":"Jo\u00e3o","family":"Marcos do \u00d3","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidade Federal da Para\u00edba , 58051-900 , Jo\u00e3o Pessoa , Brazil"}]},{"given":"Esteban","family":"da Silva","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidade Federal do Rio Grande do Norte , 59078-970 , Natal , Brazil"}]},{"given":"Evelina","family":"Shamarova","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidade Federal da Para\u00edba , 58051-900 , Jo\u00e3o Pessoa , Brazil"}]}],"member":"374","published-online":{"date-parts":[[2023,8,24]]},"reference":[{"key":"2025120518131932230_j_anona-2023-0102_ref_001","doi-asserted-by":"crossref","unstructured":"D. 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Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, UK, 1980."}],"container-title":["Advances in Nonlinear Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/anona-2023-0102\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/anona-2023-0102\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T18:41:25Z","timestamp":1764960085000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/anona-2023-0102\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,1]]},"references-count":21,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,8,1]]},"published-print":{"date-parts":[[2023,8,1]]}},"alternative-id":["10.1515\/anona-2023-0102"],"URL":"https:\/\/doi.org\/10.1515\/anona-2023-0102","relation":{},"ISSN":["2191-950X"],"issn-type":[{"type":"electronic","value":"2191-950X"}],"subject":[],"published":{"date-parts":[[2023,1,1]]},"article-number":"20230102"}}