Global bifurcation result for the p-biharmonic operator
We prove that the nonlinear eigenvalue problem for the p-biharmonic operator with $p > 1$, and $Omega$ a bounded domain in $mathbb{R}^N$ with smooth boundary, has principal positive eigenvalue $lambda_1$ which is simple and isolated. The corresponding eigenfunction is positive in $Omega$ and sati...
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Texas State University
2001-07-01
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Online Access: | http://ejde.math.txstate.edu/Volumes/2001/48/abstr.html |
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doaj-f72cf66b846f47909a2b5a2f5d8c22322020-11-24T20:46:14ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912001-07-01200148119Global bifurcation result for the p-biharmonic operatorPavel DrabekMitsuharu OtaniWe prove that the nonlinear eigenvalue problem for the p-biharmonic operator with $p > 1$, and $Omega$ a bounded domain in $mathbb{R}^N$ with smooth boundary, has principal positive eigenvalue $lambda_1$ which is simple and isolated. The corresponding eigenfunction is positive in $Omega$ and satisfies $frac{partial u}{partial n} < 0$ on $partial Omega$, $Delta u_1 < 0$ in $Omega$. We also prove that $(lambda_1,0)$ is the point of global bifurcation for associated nonhomogeneous problem. In the case $N=1$ we give a description of all eigenvalues and associated eigenfunctions. Every such an eigenvalue is then the point of global bifurcation. http://ejde.math.txstate.edu/Volumes/2001/48/abstr.htmlp-biharmonic operatorprincipal eigenvalueglobal bifurcation. |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Pavel Drabek Mitsuharu Otani |
spellingShingle |
Pavel Drabek Mitsuharu Otani Global bifurcation result for the p-biharmonic operator Electronic Journal of Differential Equations p-biharmonic operator principal eigenvalue global bifurcation. |
author_facet |
Pavel Drabek Mitsuharu Otani |
author_sort |
Pavel Drabek |
title |
Global bifurcation result for the p-biharmonic operator |
title_short |
Global bifurcation result for the p-biharmonic operator |
title_full |
Global bifurcation result for the p-biharmonic operator |
title_fullStr |
Global bifurcation result for the p-biharmonic operator |
title_full_unstemmed |
Global bifurcation result for the p-biharmonic operator |
title_sort |
global bifurcation result for the p-biharmonic operator |
publisher |
Texas State University |
series |
Electronic Journal of Differential Equations |
issn |
1072-6691 |
publishDate |
2001-07-01 |
description |
We prove that the nonlinear eigenvalue problem for the p-biharmonic operator with $p > 1$, and $Omega$ a bounded domain in $mathbb{R}^N$ with smooth boundary, has principal positive eigenvalue $lambda_1$ which is simple and isolated. The corresponding eigenfunction is positive in $Omega$ and satisfies $frac{partial u}{partial n} < 0$ on $partial Omega$, $Delta u_1 < 0$ in $Omega$. We also prove that $(lambda_1,0)$ is the point of global bifurcation for associated nonhomogeneous problem. In the case $N=1$ we give a description of all eigenvalues and associated eigenfunctions. Every such an eigenvalue is then the point of global bifurcation. |
topic |
p-biharmonic operator principal eigenvalue global bifurcation. |
url |
http://ejde.math.txstate.edu/Volumes/2001/48/abstr.html |
work_keys_str_mv |
AT paveldrabek globalbifurcationresultforthepbiharmonicoperator AT mitsuharuotani globalbifurcationresultforthepbiharmonicoperator |
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