Calculus of Variations and Geometric Measure Theory

A. Hyder - S. Iula - L. Martinazzi

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

created by martinazz on 17 Jul 2018



Inserted: 17 jul 2018

Year: 2016

ArXiv: 1610.06820 PDF


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 $2m$ $$(-\Delta)m uk = Vke{2muk}\quad \text{in }\Omega, \quad \limsup{k\to\infty} \int\Omega e{2muk}dx<\infty,$$ and blowing up exactly on the set $S_{\varphi}:=\{x\in \Omega:\varphi(x)=0\}$, i.e. $$\lim{k\to\infty} uk(x)=+\infty \text{ for }x\in S{\varphi} \text{ and }\lim{k\to\infty} uk(x)=-\infty \text{ for }x\in \Omega\setminus S{\varphi},$$ 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.