SUPERCRITICAL PROBLEMS IN DOMAINS WITH THIN TOROIDAL HOLES

Abstract

In this paper we study the Lane-Emden-Fowler equation (P)ε, ∫, Δu+|q|q-2 u=0 in Dε, u=0 on ∂Dε Here Dε = D\-x ε D : dist(x, λℓ') ≤ ε}, D is a smooth bounded domain in ℝN, λ ℓ in an ℓ- dimensional closed manifold such that λ ℓ ⊂ D with 1 ≤ ℓ 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)ε increases as ε goes to zero.