Nonexistence of bounded energy solutions for a fourth order equation on thin annuli

In this paper we study the problem Pε : Δ2uε uε(n+4)/(n-4), uε > in 0 in Aε; uε = Δuε = 0 on ∂Aε, where {Aε is contained in Rn, ε > 0} is a family of bounded annulus shaped domains such that Aε becomes 'thin' as ε → 0. Our main result is the following: Assume n ≥ 6 and let C > 0 be a constant. Then there exists ε0 > 0 such that for any ε < ε0, the problem Pε has no solution uε, whose energy, ∫Aεvertical barΔuεvertical bar2 is less than C. Our proof involves a rather delicate analysis of asymptotic profiles of solutions uε when ε → 0. (author)