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

TL;DR

This paper constructs solutions to the prescribed Q-curvature equation in Euclidean space that blow up exactly on prescribed sets, demonstrating the sharpness of previous results and extending them to boundary and entire space cases.

Contribution

It introduces a method to produce blow-up solutions with prescribed singular sets for higher-order Liouville equations, extending known results to new geometric contexts.

Findings

Solutions blow up exactly on the zero set of a given function.

The blow-up solutions extend previous sharpness results.

The method applies to boundary and entire space cases.

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 $2m$ $$(-\Delta)^m u_k = V_ke^{2mu_k}\quad \text{in }\Omega, \quad \limsup_{k\to\infty} \int_\Omega e^{2mu_k}dx<\infty,$$ and blowing up exactly on the set $S_{\varphi}:=\{x\in \Omega:\varphi(x)=0\}$, i.e. $$\lim_{k\to\infty} u_k(x)=+\infty \text{ for }x\in S_{\varphi} \text{ and }\lim_{k\to\infty} u_k(x)=-\infty \text{ for }x\in \Omega\setminus S_{\varphi},$$ 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.