Existence of two solutions for the Bahri–Coron problem in an annular domain with a thin hole

We will show that the problem−Δu=|u|4/(N−2)uin Ω,u=0on ∂Ω has at least two pairs of solutions if Ω   is an annular domain with a thin hole in RNRN (N⩾3N⩾3).