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

M. Ben Ayed; Abdelbaki Selmi

doi:10.3934/cpaa.2010.9.1705

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2010, Volume 9, Issue 6: 1705-1722. Doi: 10.3934/cpaa.2010.9.1705

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

M. Ben Ayed^1, and
Abdelbaki Selmi^2,

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

Received: July 31, 2009

Revised: March 31, 2010

Published: October 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J60, 35J65, 58E05.

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