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

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 $k\leq \frac{n-2m}{2}$.