Single blow-up solutions for a slightly subcritical biharmonic equation

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.