Large blow-up sets for the prescribed Q -curvature equation in the Euclidean space

Let m ≥ 2 be an integer. For any open domain Ω ⊂ R2m, non-positive function φ ∈ C∞(Ω) such that ∆mφ ≡ 0, and bounded sequence (Vk) ⊂ L∞(Ω) we prove the existence of a sequence of functions (uk) ⊂ C2m−1(Ω) solving the Liouville equation of order 2m Z (−∆)muk = Vke2muk in Ω, lim sup e2muk dx < ∞, k→∞ Ω andblowingupexactlyonthesetSφ :={x∈Ω:φ(x)=0},i.e. lim uk(x)=+∞forx∈Sφ and lim uk(x)=−∞forx∈ΩSφ, k→∞ k→∞ thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this 2m result to the boundary of Ω and to the case Ω = R open. . Several related problems remain