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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Texas State University
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doaj-c5ac1bad2a194f7cb7a056acd87fed992020-11-24T22:57:24ZengTexas 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 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Evandro Monteiro |
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 |
author_facet |
Evandro Monteiro |
author_sort |
Evandro Monteiro |
title |
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_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_sort |
asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues |
publisher |
Texas State University |
series |
Electronic Journal of Differential Equations |
issn |
1072-6691 |
publishDate |
2011-11-01 |
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. |
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 |
_version_ |
1725650891981193216 |