Nontrivial solutions for perturbations of a Hardy–Sobolev operator on unbounded domains

In this paper we study the existence of nontrivial solution of the problem −Δpu−(μ/[d(x)]p)× |u|p−2u = f (u) in Ω and u = 0 on ∂Ω, where Ω = Ω × RN−k,Ω is a bounded domain with smooth boundary in Rk, 1 < k