Boundary clustered layers near the higher critical exponents

Abstract

We consider the supercritical problem -Delta u = vertical bar u vertical bar(p-2)u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N and p smaller than the critical exponent 2(N,k)* := 2(N-k)/N-k-2 for the Sobolev embedding of H-1(RN-k) in L-q(RN-k), 1 <= k <= N - 3. We show that in some suitable domains Omega there are positive and sign changing solutions with positive and negative layers which concentrate along one or several k-dimensional submanifolds of partial derivative Omega as p approaches 2(N,k)* from below. (C) 2013 Elsevier Inc. All rights reserved