Sign-changing solutions at the almost Hénon critical exponent

We study the problem(Pα)−Δu=|x|α|u|4+2αN−2−εu   in Ω,u=0   on ∂Ω, where Ω is a bounded smooth domain in RN, N≥3, which is symmetric with respect to x1,x2,…,xN and contains the origin, α>0, and ε>0 is a small parameter. We construct solutions to (Pα) with the shape of a sign-changing tower of bubbles of order α that concentrate and blow-up at the origin as ε→0. We also study a slightly Hénon supercritical dual version of (Pα) in an exterior domain, for which we found solutions with the shape of a flat sign-changing tower of bubbles of order α that disappear as ε→0.