Boundary towers of layers for some supercritical problems

Abstract

We consider the supercritical problem -Delta u = vertical bar u vertical bar(p-1)u in D, u = 0 on partial derivative D, where D is a bounded smooth domain in RN and p is smaller than the kappa-th critical Sobolev exponent 2*(N,kappa) := N-kappa+2/N-kappa-2 with 1 <= kappa <= N - 3. We show that in some suitable torus-like domains D there exists an arbitrary large number of sign-changing solutions with alternate positive and negative layers which concentrate at different rates along a kappa-dimensional submanifold of partial derivative D as p approaches 2*(N,kappa) from below.