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

Let m≥2 be an integer. For any open domain Ω⊂R2m, non-positive function φ∈C∞(Ω) such that Δmφ≡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 (−Δ)muk=Vke2mukin Ω,limsupk→∞∫Ωe2mukdx<∞,
and blowing up exactly on the set Sφ:={x∈Ω:φ(x)=0}, i.e.
limk→∞uk(x)=+∞ for x∈Sφandlimk→∞uk(x)=−∞ for x∈Ω∖Sφ,
thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of Ω and to the case Ω=R2m. Several related problems remain open.