Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS
We study global bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem arising in MEMS {−(u′(x)1+(u′(x))2)′=λ(1−u)p,u
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| Veröffentlicht in: | Nonlinear analysis 2013-09, Vol.89, p.284-298 |
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| container_title | Nonlinear analysis |
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| creator | Cheng, Yan-Hsiou Hung, Kuo-Chih Wang, Shin-Hwa |
| description | We study global bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem arising in MEMS {−(u′(x)1+(u′(x))2)′=λ(1−u)p,u |
| doi_str_mv | 10.1016/j.na.2013.04.018 |
| format | Article |
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We determine the exact number of positive solutions by the values of p,L and λ. Moreover, for p≥1, the bifurcation diagram undergoes fold and splitting bifurcations. While for 0<p<1, the bifurcation diagram undergoes fold, splitting and segment-shrinking bifurcations. Our results extend and improve those of Brubaker and Pelesko [N.D. Brubaker, J.A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012) 5086–5102] and Pan and Xing [H. Pan, R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to a MEMS model, Nonlinear Anal. RWA 13 (2012) 2432–2445] by generalizing the nonlinearity (1−u)−2 to (1−u)−p with general p∈(1,∞). We also answer an open question raised by Brubaker and Pelesko on the extension of (global) bifurcation diagram results to general p>0. Concerning this open question, we find and prove that global bifurcation diagrams for 0<p<1 are different to and more complicated than those for p≥1.]]></description><identifier>ISSN: 0362-546X</identifier><identifier>EISSN: 1873-5215</identifier><identifier>DOI: 10.1016/j.na.2013.04.018</identifier><language>eng</language><publisher>Elsevier Ltd</publisher><subject>Bifurcations ; Exact multiplicity ; Global bifurcation diagram ; MEMS ; Positive solution ; Prescribed mean curvature problem</subject><ispartof>Nonlinear analysis, 2013-09, Vol.89, p.284-298</ispartof><rights>2013 Elsevier Ltd</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c426t-ba11fb89ee128091c91a6d1aada352e738bf129d4cf0dde1e0d384dd0f1252073</citedby><cites>FETCH-LOGICAL-c426t-ba11fb89ee128091c91a6d1aada352e738bf129d4cf0dde1e0d384dd0f1252073</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.na.2013.04.018$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Cheng, Yan-Hsiou</creatorcontrib><creatorcontrib>Hung, Kuo-Chih</creatorcontrib><creatorcontrib>Wang, Shin-Hwa</creatorcontrib><title>Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS</title><title>Nonlinear analysis</title><description><![CDATA[We study global bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem arising in MEMS {−(u′(x)1+(u′(x))2)′=λ(1−u)p,u<1,−L<x<L,u(−L)=u(L)=0, where λ>0 is a bifurcation parameter, and p,L>0 are two evolution parameters. We determine the exact number of positive solutions by the values of p,L and λ. Moreover, for p≥1, the bifurcation diagram undergoes fold and splitting bifurcations. While for 0<p<1, the bifurcation diagram undergoes fold, splitting and segment-shrinking bifurcations. Our results extend and improve those of Brubaker and Pelesko [N.D. Brubaker, J.A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012) 5086–5102] and Pan and Xing [H. Pan, R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to a MEMS model, Nonlinear Anal. RWA 13 (2012) 2432–2445] by generalizing the nonlinearity (1−u)−2 to (1−u)−p with general p∈(1,∞). We also answer an open question raised by Brubaker and Pelesko on the extension of (global) bifurcation diagram results to general p>0. Concerning this open question, we find and prove that global bifurcation diagrams for 0<p<1 are different to and more complicated than those for p≥1.]]></description><subject>Bifurcations</subject><subject>Exact multiplicity</subject><subject>Global bifurcation diagram</subject><subject>MEMS</subject><subject>Positive solution</subject><subject>Prescribed mean curvature problem</subject><issn>0362-546X</issn><issn>1873-5215</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNqFkUGL1EAQhYMoOK7ePfbRS2JVJ51kvMmyrsIuHlTw1lS6K0sNSffYnQzuv_Anm3G8iqeCV-97VPGK4jVChYDt20MVqNKAdQVNBdg_KXbYd3VpNJqnxQ7qVpemab8_L17kfAAA7Op2V_y6neJAkxpkXJOjRWJQXugh0ZwVBa_4J7lFzeu0yHESJ8ujiqM6xiyLnFjlOK1nKKsxJkUqBi69zBzyJm65x8TZJRnYq5kpKLemEy1r4m0Th4lnRUmyhAclQd3f3H95WTwbacr86u-8Kr59uPl6_bG8-3z76fr9Xeka3S7lQIjj0O-ZUfewR7dHaj0SeaqN5q7uhxH13jduBO8ZGXzdN97DphoNXX1VvLnkbnf8WDkvdpbseJoocFyzRQNt3Ruzb_9vbTpjtNbYb1a4WF2KOSce7THJTOnRIthzT_ZgA9lzTxYaC3-QdxeEt29PwslmJxwce0nsFuuj_Bv-DfAinXU</recordid><startdate>20130901</startdate><enddate>20130901</enddate><creator>Cheng, Yan-Hsiou</creator><creator>Hung, Kuo-Chih</creator><creator>Wang, Shin-Hwa</creator><general>Elsevier Ltd</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7SP</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20130901</creationdate><title>Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS</title><author>Cheng, Yan-Hsiou ; Hung, Kuo-Chih ; Wang, Shin-Hwa</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c426t-ba11fb89ee128091c91a6d1aada352e738bf129d4cf0dde1e0d384dd0f1252073</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><topic>Bifurcations</topic><topic>Exact multiplicity</topic><topic>Global bifurcation diagram</topic><topic>MEMS</topic><topic>Positive solution</topic><topic>Prescribed mean curvature problem</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cheng, Yan-Hsiou</creatorcontrib><creatorcontrib>Hung, Kuo-Chih</creatorcontrib><creatorcontrib>Wang, Shin-Hwa</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Electronics & Communications Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Nonlinear analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cheng, Yan-Hsiou</au><au>Hung, Kuo-Chih</au><au>Wang, Shin-Hwa</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS</atitle><jtitle>Nonlinear analysis</jtitle><date>2013-09-01</date><risdate>2013</risdate><volume>89</volume><spage>284</spage><epage>298</epage><pages>284-298</pages><issn>0362-546X</issn><eissn>1873-5215</eissn><abstract><![CDATA[We study global bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem arising in MEMS {−(u′(x)1+(u′(x))2)′=λ(1−u)p,u<1,−L<x<L,u(−L)=u(L)=0, where λ>0 is a bifurcation parameter, and p,L>0 are two evolution parameters. We determine the exact number of positive solutions by the values of p,L and λ. Moreover, for p≥1, the bifurcation diagram undergoes fold and splitting bifurcations. While for 0<p<1, the bifurcation diagram undergoes fold, splitting and segment-shrinking bifurcations. Our results extend and improve those of Brubaker and Pelesko [N.D. Brubaker, J.A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012) 5086–5102] and Pan and Xing [H. Pan, R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to a MEMS model, Nonlinear Anal. RWA 13 (2012) 2432–2445] by generalizing the nonlinearity (1−u)−2 to (1−u)−p with general p∈(1,∞). We also answer an open question raised by Brubaker and Pelesko on the extension of (global) bifurcation diagram results to general p>0. Concerning this open question, we find and prove that global bifurcation diagrams for 0<p<1 are different to and more complicated than those for p≥1.]]></abstract><pub>Elsevier Ltd</pub><doi>10.1016/j.na.2013.04.018</doi><tpages>15</tpages></addata></record> |
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| subjects | Bifurcations Exact multiplicity Global bifurcation diagram MEMS Positive solution Prescribed mean curvature problem |
| title | Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS |
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