Local behavior of positive solutions of higher order conformally invariant equations with a singular set

TL;DR

This paper investigates the local behavior, asymptotic symmetry, and global symmetry of positive solutions to higher order conformally invariant equations with singular sets, establishing estimates and symmetry properties under geometric conditions.

Contribution

It provides new asymptotic blow-up rate estimates and symmetry results for solutions near singular sets of specific dimensions in higher order conformally invariant equations.

Findings

Established blow-up rate estimates near singular sets with controlled Minkowski dimension.

Proved asymptotic symmetry of solutions near smooth singular manifolds.

Derived global symmetry results when the singular set is a hyperplane.

Abstract

We study some properties of positive solutions to the higher order conformally invariant equation with a singular set $$ (-\Delta)^m u = u^{\frac{n+2m}{n-2m}} ~~~~~~ \textmd{in} ~ \Omega \backslash \Lambda, $$ where $\Omega \subset \mathbb{R}^n$ is an open domain, $\Lambda$ is a closed subset of $\mathbb{R}^n$, $1 \leq m < n/2$ and $m$ is an integer. We first establish an asymptotic blow up rate estimate for positive solutions near the singular set $\Lambda$ when $\Lambda \subset \Omega$ is a compact set with the upper Minkowski dimension $\overline{\textmd{dim}}_M(\Lambda) < \frac{n-2m}{2}$, or is a smooth $k$-dimensional closed manifold with $k\leq \frac{n-2m}{2}$. We also show the asymptotic symmetry of singular positive solutions suppose $\Lambda \subset \Omega$ is a smooth $k$-dimensional closed manifold with $k\leq \frac{n-2m}{2}$. Finally, a global symmetry result for solutions is obtained when $\Omega$ is the whole space and $\Lambda$ is a $k$-dimensional hyperplane with $k\leq \frac{n-2m}{2}$.