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.