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

Abstract

Let $m\ge 2$ be an integer. For any open domain $\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\in C^{\infty}(\Omega)$ such that $\Delta^m \varphi \equiv 0$ and bounded sequence $(V_k) \subset L^\infty (\Omega)$ we prove the existence of a sequence of functions $(u_k) \subset C^{2m-1}(\Omega)$ solving the Liouville equation of order 2m $$(-\Delta)^m u_k = V_k e^{2m u_k} in \Omega, \limsup_{k\to\infty} \int_{\Omega} e^{2m u_k} dx < \infty,$$ and blowing up exactly on the set $S_{\varphi} := \{x\in\Omega : \varphi(x) = 0\}$, i.e. $$\lim_{k\to\infty} u_k(x) = +\infty for x\in S_{\varphi} and \lim_{k\to\infty} u_k(x) = -\infty for x\in\Omega\setminus S_{\varphi},$$ thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of $\Omega$ and to the case $\Omega = \mathbb{R}^{2m}$. Several related problems remain open