Concentration phenomena for fourth-order elliptic equations with critical exponent

Abstract

We consider the nonlinear equation Δ2u = u n+4/n-4 - εu with u > 0 in Ω and u = Δu = 0 on ∂Ω. Where Ω is a smooth bounded domain in ℝn, n ≥ 9, and ε is a small positive parameter. We study the existence of solutions which concentrate around one or two points of Ω. We show that this problem has no solutions that concentrate around a point of Ω as ε approaches 0. In contract to this, we construct a domain for which there exists a family of solutions which blow-up and concentrate in two different points of Ω as ε approaches 0.