Boundary clustered layers near the higher critical exponents

We consider the supercritical problem−Δu=|u|p−2uin Ω,u=0on ∂Ω, where Ω   is a bounded smooth domain in RNRN and p   smaller than the critical exponent 2N,k⁎:=2(N−k)N−k−2 for the Sobolev embedding of H1(RN−k)H1(RN−k) in Lq(RN−k)Lq(RN−k), 1⩽k⩽N−31⩽k⩽N−3. We show that in some suitable domains Ω there are positive and sign changing solutions with positive and negative layers which concentrate along one or several k-dimensional submanifolds of ∂Ω as p   approaches 2N,k⁎ from below.