Single blow-up solutions for a slightly subcritical biharmonic equation

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent ( P ε ): ∆ 2 u = u 9 − ε , u > 0 in Ω and u = ∆ u = 0 on ∂ Ω , where Ω is a smooth bounded domain in ℝ 5 , ε > 0 . We study the asymptotic behavior of solutions of ( P ε ) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x 0 ∈ Ω as ε → 0 , moreover x 0 is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point x 0 of the Robin's function, there exist solutions of ( P ε ) concentrating around x 0 as ε → 0 .