Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space
Let $m\ge 2$ be an integer. For any open domain $Ω\subset\mathbb{R}^{2m}$, non-positive function $φ\in C^\infty(Ω)$ such that $Δ^m φ\equiv 0$, and bounded sequence $(V_k)\subset L^\infty(Ω)$ we prove the existence of a sequence of functions $(u_k)\subset C^{2m-1}(Ω)$ solving the Liouville equation of order $2m$ $$(-Δ)^m u_k = V_ke^{2mu_k}\quad \text{in }Ω, \quad \limsup_{k\to\infty} \int_Ωe^{2mu_k}dx<\infty,$$ and blowing up exactly on the set $S_φ:=\{x\in Ω:φ(x)=0\}$, i.e. $$\lim_{k\to\infty} u_k(x)=+\infty \text{ for }x\in S_φ \text{ and }\lim_{k\to\infty} u_k(x)=-\infty \text{ for }x\in Ω\setminus S_φ,$$ 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 $Ω=\mathbb{R}^{2m}$. Several related problems remain open.