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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Main Author: Evandro Monteiro
Format: Article
Language:English
Published: Texas State University 2011-11-01
Series:Electronic Journal of Differential Equations
Subjects:
Online Access:http://ejde.math.txstate.edu/Volumes/2011/145/abstr.html
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spelling 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
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