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 $\phi \in C^{\infty }(\Omega )$ such that ${\Delta }^m\phi \equiv 0$, 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 $2m$ $$(-\Delta {)}^m{u}_k=V_k{e}^{2m{u}_k}\text{in }\Omega ,\underset{k\to \infty }{\mathrm{limsup}}{\int }_{\Omega }{e}^{2m{u}_k}dx<\infty ,$$ and blowing up exactly on the set $S_{\phi }:=\{x\in \Omega :\phi (x)=0\}$, i.e. $$\underset{k\to \infty }{\lim }{u}_k(x)=+\infty \text{ for }x\in S_{\phi }\text{ and }\underset{k\to \infty }{\lim }{u}_k(x)=-\infty \text{ for }x\in \Omega \setminus S_{\phi },$$ 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.

References in corpus (2)

• Concentration-compactness phenomena in the higher order Liouville's equation
• Quantization for the prescribed Q-curvature equation on open domains

Cited by in corpus (1)

• Concentration phenomena for the fractional $Q$-curvature equation in dimension 3 and fractional Poisson formulas