Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues
In this article we prove the existence of multiple solutions for the fourth-order elliptic problem $$displaylines{ Delta^2u+cDelta u = g(x,u) quadhbox{in } Omegacr u =Delta u= 0 quadhbox{on } partial Omega, }$$ where $Omega subset mathbb{R}^N$ is a bounded domain, $g:Omegaimesmathbb{R}o mathb...
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| Format: | Article |
| Language: | English |
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Texas State University
2011-11-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2011/145/abstr.html |
| _version_ | 1831569428207108096 |
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| author | Evandro Monteiro |
| author_facet | Evandro Monteiro |
| author_sort | Evandro Monteiro |
| collection | DOAJ |
| description | In this article we prove the existence of multiple solutions for the fourth-order elliptic problem $$displaylines{ Delta^2u+cDelta u = g(x,u) quadhbox{in } Omegacr u =Delta u= 0 quadhbox{on } partial Omega, }$$ where $Omega subset mathbb{R}^N$ is a bounded domain, $g:Omegaimesmathbb{R}o mathbb{R}$ is a function of class $C^1$ such that $g(x,0)=0$ and it is asymptotically linear at infinity. We study the cases when the parameter c is less than the first eigenvalue, and between two consecutive eigenvalues of the Laplacian. To obtain solutions we use the Saddle Point Theorem, the Linking Theorem, and Critical Groups Theory. |
| first_indexed | 2024-12-17T12:34:07Z |
| format | Article |
| id | doaj.art-c5ac1bad2a194f7cb7a056acd87fed99 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-17T12:34:07Z |
| publishDate | 2011-11-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-c5ac1bad2a194f7cb7a056acd87fed992022-12-21T21:48:26ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912011-11-012011145,111Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvaluesEvandro MonteiroIn this article we prove the existence of multiple solutions for the fourth-order elliptic problem $$displaylines{ Delta^2u+cDelta u = g(x,u) quadhbox{in } Omegacr u =Delta u= 0 quadhbox{on } partial Omega, }$$ where $Omega subset mathbb{R}^N$ is a bounded domain, $g:Omegaimesmathbb{R}o mathbb{R}$ is a function of class $C^1$ such that $g(x,0)=0$ and it is asymptotically linear at infinity. We study the cases when the parameter c is less than the first eigenvalue, and between two consecutive eigenvalues of the Laplacian. To obtain solutions we use the Saddle Point Theorem, the Linking Theorem, and Critical Groups Theory.http://ejde.math.txstate.edu/Volumes/2011/145/abstr.htmlAsymptotically linearMorse theoryshifting theoremmultiplicity of solutions |
| spellingShingle | Evandro Monteiro Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues Electronic Journal of Differential Equations Asymptotically linear Morse theory shifting theorem multiplicity of solutions |
| title | Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues |
| title_full | Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues |
| title_fullStr | Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues |
| title_full_unstemmed | Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues |
| title_short | Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues |
| title_sort | asymptotically linear fourth order elliptic problems whose nonlinearity crosses several eigenvalues |
| topic | Asymptotically linear Morse theory shifting theorem multiplicity of solutions |
| url | http://ejde.math.txstate.edu/Volumes/2011/145/abstr.html |
| work_keys_str_mv | AT evandromonteiro asymptoticallylinearfourthorderellipticproblemswhosenonlinearitycrossesseveraleigenvalues |