Sign-changing solutions at the almost Henon critical exponent

We study the problem -Delta u = vertical bar x vertical bar(alpha)vertical bar u vertical bar (4+2 alpha/N-2-epsilon) u in Omega, u=0 on partial derivative Omega, (p(alpha)) where Omega is a bounded smooth domain in R-N, N >= 3, which is symmetric with respect to x(1), x(2),...,x(N) and contains the origin, alpha > 0, and epsilon > 0 is a small parameter. We construct solutions to (P-alpha) with the shape of a sign-changing tower of bubbles of order a that concentrate and blow-up at the origin as epsilon -> 0. We also study a slightly Henon supercritical dual version of (P-alpha) in an exterior domain, for which we found solutions with the shape of a flat sign-changing tower of bubbles of order alpha that disappear as epsilon -> 0. (C) 2018 Elsevier Inc. All rights reserved.