Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 121, pp. 1-22.
Title: Concentration phenomena for fourth-order elliptic
equations with critical exponent
Author: Mokhless Hammami (Faculte des Sciences de Sfax, Tunisia)
Abstract:
We consider the nonlinear equation
$$
\Delta ^2u= u^{\frac{n+4}{n-4}}-\varepsilon u
$$
with $u>0$ 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.