Supercritical problems in domains with thin toroidal holes

Abstract

In this paper we study the Lane-Emden-Fowler equation (P)(epsilon) {(Delta)u+vertical bar u vertical bar(q-2) u = 0 in D-epsilon,D- u = 0 on partial derivative D-epsilon. Here D-c = D\ {x epsilon D : dist (x, Gamma(l) ) <= epsilon }, D is a smooth bounded domain in R-N, Gamma(l) is an l-dimensional closed manifold such that Gamma l subset of D with 1 <= l <= N - 3 and q = 2(N - l)/ N-l-2. We prove that, under some symmetry assumptions, the number of sign changing solutions to (P)(epsilon), increases as goes to zero.