Abstract and Applied Analysis
Volume 2006 (2006), Article ID 18387, 20 pages
doi:10.1155/AAA/2006/18387

Single blow-up solutions for a slightly subcritical biharmonic equation

Khalil El Mehdi1,2

1Faculté des Sciences et Techniques, Université de Nouakchott, Nouakchott BP 5026, Mauritania
2The Abdus Salam ICTP, Trieste 34014, Italy

Received 29 October 2004; Accepted 20 January 2005

Copyright © 2006 Khalil El Mehdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (Pε): ∆2u=u9−ε, u>0 in Ω and u=∆u=0 on ∂Ω, where Ω is a smooth bounded domain in ℝ5, ε>0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x0∈Ω as ε→0, moreover x0 is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point x0 of the Robin's function, there exist solutions of (Pε) concentrating around x0 as ε→0.