Boundary towers of layers for some supercritical problems

We consider the supercritical problem−Δu=|u|p−1uin D,u=0on ∂D, where DD is a bounded smooth domain in RNRN and p is smaller than the κ  -th critical Sobolev exponent 2N,κ⁎:=N−κ+2N−κ−2 with 1⩽κ⩽N−31⩽κ⩽N−3. We show that in some suitable torus-like domains DD there exists an arbitrary large number of sign-changing solutions with alternate positive and negative layers which concentrate at different rates along a κ  -dimensional submanifold of ∂D∂D as p   approaches 2N,κ⁎ from below.