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 − Δ u = | u | p − 1 u i n D , u = 0 o n ∂ D , where 𝔇 is a smooth open bounded domain in ℝn 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 → + ∞.