Concentration phenomena for fourth-order elliptic equations with critical exponent
We consider the nonlinear equation $$ Delta ^2u= u^{frac{n+4}{n-4}}-varepsilon u $$ with $u$ greater than 0 in $Omega$ and $u=Delta u=0$ on $partialOmega$. Where $Omega$ is a smooth bounded domain in $mathbb{R}^n$, $ngeq 9$, and $varepsilon$ is a small positive parameter. We study the existence of...
Main Author: | |
---|---|
Format: | Article |
Language: | English |
Published: |
Texas State University
2004-10-01
|
Series: | Electronic Journal of Differential Equations |
Subjects: | |
Online Access: | http://ejde.math.txstate.edu/Volumes/2004/121/abstr.html |
id |
doaj-f1e23779f0a847ad8fadafe677846f57 |
---|---|
record_format |
Article |
spelling |
doaj-f1e23779f0a847ad8fadafe677846f572020-11-25T00:23:44ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912004-10-012004121122Concentration phenomena for fourth-order elliptic equations with critical exponentMokhless HammamiWe consider the nonlinear equation $$ Delta ^2u= u^{frac{n+4}{n-4}}-varepsilon u $$ with $u$ greater than 0 in $Omega$ and $u=Delta u=0$ on $partialOmega$. Where $Omega$ is a smooth bounded domain in $mathbb{R}^n$, $ngeq 9$, and $varepsilon$ is a small positive parameter. We study the existence of solutions which concentrate around one or two points of $Omega$. We show that this problem has no solutions that concentrate around a point of $Omega$ as $varepsilon$ approaches 0. In contrast to this, we construct a domain for which there exists a family of solutions which blow-up and concentrate in two different points of $Omega$ as $varepsilon$ approaches 0.http://ejde.math.txstate.edu/Volumes/2004/121/abstr.htmlFourth order elliptic equationscritical Sobolev exponentblowup solution. |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Mokhless Hammami |
spellingShingle |
Mokhless Hammami Concentration phenomena for fourth-order elliptic equations with critical exponent Electronic Journal of Differential Equations Fourth order elliptic equations critical Sobolev exponent blowup solution. |
author_facet |
Mokhless Hammami |
author_sort |
Mokhless Hammami |
title |
Concentration phenomena for fourth-order elliptic equations with critical exponent |
title_short |
Concentration phenomena for fourth-order elliptic equations with critical exponent |
title_full |
Concentration phenomena for fourth-order elliptic equations with critical exponent |
title_fullStr |
Concentration phenomena for fourth-order elliptic equations with critical exponent |
title_full_unstemmed |
Concentration phenomena for fourth-order elliptic equations with critical exponent |
title_sort |
concentration phenomena for fourth-order elliptic equations with critical exponent |
publisher |
Texas State University |
series |
Electronic Journal of Differential Equations |
issn |
1072-6691 |
publishDate |
2004-10-01 |
description |
We consider the nonlinear equation $$ Delta ^2u= u^{frac{n+4}{n-4}}-varepsilon u $$ with $u$ greater than 0 in $Omega$ and $u=Delta u=0$ on $partialOmega$. Where $Omega$ is a smooth bounded domain in $mathbb{R}^n$, $ngeq 9$, and $varepsilon$ is a small positive parameter. We study the existence of solutions which concentrate around one or two points of $Omega$. We show that this problem has no solutions that concentrate around a point of $Omega$ as $varepsilon$ approaches 0. In contrast to this, we construct a domain for which there exists a family of solutions which blow-up and concentrate in two different points of $Omega$ as $varepsilon$ approaches 0. |
topic |
Fourth order elliptic equations critical Sobolev exponent blowup solution. |
url |
http://ejde.math.txstate.edu/Volumes/2004/121/abstr.html |
work_keys_str_mv |
AT mokhlesshammami concentrationphenomenaforfourthorderellipticequationswithcriticalexponent |
_version_ |
1725355306605608960 |