# American Institute of Mathematical Sciences

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

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.
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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