Blowing up solutions for a biharmonic equation with critical nonlinearity

Article type: Research Article

Authors: El Mehdi, Khalil | Hammami, Mokhless

Affiliations: Faculté des Sciences et Techniques, Université de Nouakchott, BP 5026, Nouakchott, Mauritania and The Abdus Salam ICTP, Trieste, Italy E-mail: khalil@univ-nkc.mr | Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia E-mail: Mokhless.Hammami@fss.rnu.tn

Abstract: In this paper we consider the following biharmonic equation with critical exponent \[$(P_{\varepsilon})\dvt\Delta^{2}u=Ku^{\frac{n+4}{n-4}-\varepsilon}$, u>0 in Ω and u=Δu=0 on \[$\curpartial \varOmega $, where Ω is a smooth bounded domain in \[$\mathbb{R}^{n}$, n≥5, ε is a small positive parameter, and K is a smooth positive function in \[$\overline{ \varOmega }$. We construct solutions of (Pε) which blow up and concentrate at strict local maximum of K either at the boundary or in the interior of Ω. We also construct solutions of (Pε) concentrating at an interior strict local minimum point of K. Finally, we prove a nonexistence result for the corresponding supercritical problem which is in sharp contrast to what happened for (Pε).

Keywords: fourth-order elliptic equations, critical Sobolev exponent, biharmonic operator

Journal: Asymptotic Analysis, vol. 45, no. 3-4, pp. 191-225, 2005