Electron. J. Diff. Eqns., Vol. 2004(2004), No. 121, pp. 1-22.

Concentration phenomena for fourth-order elliptic equations with critical exponent

Mokhless Hammami

Abstract:
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 $\partial\Omega$. Where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\geq 9$, 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.

Submitted August 25, 2004. Published October 14, 2004.
Math Subject Classifications: 35J65, 35J40, 58E05.
Key Words: Fourth order elliptic equations; critical Sobolev exponent; blowup solution.

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Mokhless Hammami
Département de Mathématiques
Faculté des Sciences de Sfax
Route Soukra, 3018, Sfax, Tunisia
email: Mokhless.Hammami@fss.rnu.tn

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