Concentration phenomena for fourth-order elliptic equations with critical exponent

Hammami, Mokhless

Electronic Journal of Differential Equations, Tome 2004 (2004).

Résumé

Summary: We consider the nonlinear equation $$ \Delta ^2u= u^{\frac{n+4}{n-4}}-\varepsilon u $$ with $u$ 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.

Détail

Classification : 35J65, 35J40, 58E05
Keywords: fourth order elliptic equations, critical Sobolev exponent, blowup solution