Critical polyharmonic problems with singular nonlinearities

Let us consider the Dirichlet problem {(−Δ)mu=|u|pα−2u|x|α+λuinΩDβu|∂Ω=0for|β|≤m−1 where Ω⊂RnΩ⊂Rn is a bounded open set containing the origin, n>2mn>2m, 0<α<2m0<α<2m and pα=2(n−α)/(n−2m)pα=2(n−α)/(n−2m). We find that, when n≥4mn≥4m, this problem has a solution for any 0<λ<Λm,10<λ<Λm,1, where Λm,1Λm,1 is the first Dirichlet eigenvalue of (−Δ)m(−Δ)m in ΩΩ, while, when 2m