Clustered boundary layer sign-changing solutions for a supercritical problem

Abstract

We study the existence and profile of sign-changing solutions of the supercritical problem -Delta u = |u|(p-1)u in D, u = 0 on partial derivative D, where D is a smooth open bounded domain in R-n and p > 1. In particular, for suitable domains D, we prove that, for any integer m, if p is large enough, such a problem has a sign-changing solution which concentrates positively and negatively along m different (n - 2)-dimensional submanifolds of the boundary of D that collapse to a suitable submanifold of the boundary as p -> + infinity.