Concentration phenomena for fourth-order elliptic equations with critical exponent

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 $partialOmega$. Where $Omega$ is a smooth bounded domain in $mathbb{R}^n$, $ngeq 9$, 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.

Keywords

• Fourth order elliptic equations
• critical Sobolev exponent
• blowup solution.