ar X iv : m at h / 04 01 06 1 v 1 [ m at h . A P ] 7 J an 2 00 4 On a Biharmonic Equation Involving Nearly Critical Exponent ∗

Abstract. This paper is concerned with a biharmonic equation under the Navier boundary condition (P∓ε) : ∆u = u n+4 n−4 , u > 0 in Ω and u = ∆u = 0 on ∂Ω, where Ω is a smooth bounded domain in R, n ≥ 5, and 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 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 (P−ε) concentrating around x0 as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P−ε), the supercritical problem (P+ε) has no solutions which concentrate around a point of Ω as ε → 0.