Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain

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November 2010, 9(6): 1705-1722. doi: 10.3934/cpaa.2010.9.1705

Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain

1.	Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia
2.	Département de Mathématiques, Faculté des Sciences de Bizerte, Zarzouna 7021, Bizerte, Tunisia

Received August 2009 Revised April 2010 Published August 2010

Abstract
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In this paper, we consider the problem $(Q_\varepsilon)$ : $\Delta ^2 u= u^9 +\varepsilon f(x)$ in $\Omega$, $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a bounded and smooth domain in $R^5$, $\varepsilon$ is a small positive parameter, and $f$ is a smooth function. Our main purpose is to characterize the solutions with some assumptions on the energy. We prove that these solutions blow up at a critical point of a function depending on $f$ and the regular part of the Green's function. Moreover, we construct families of solutions of $(Q_\varepsilon)$ which satisfy the conclusions of the first part.

Keywords: critical Sobolev exponent, Fourth order elliptic equation, Biharmonic operator..

Mathematics Subject Classification: Primary: 35J60, 35J65, 58E0.

Citation: M. Ben Ayed, Abdelbaki Selmi. Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1705-1722. doi: 10.3934/cpaa.2010.9.1705

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