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 $$(-\Delta {)}^{m}u=u^{\frac{n+2m}{n-2m}}\text{in}\Omega \backslash \Lambda ,$$ where $\Omega \subset {\mathbb{R}}^n$ is an open domain, $\Lambda$ is a closed subset of ${\mathbb{R}}^n$, $1\le m<n/2$ 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 ${\overline{\text{dim}}}_M(\Lambda )<\frac{n-2m}{2}$, or is a smooth $k$-dimensional closed manifold with $k\le \frac{n-2m}{2}$. We also show the asymptotic symmetry of singular positive solutions suppose $\Lambda \subset \Omega$ is a smooth $k$-dimensional closed manifold with $k\le \frac{n-2m}{2}$. Finally, a global symmetry result for solutions is obtained when $\Omega$ is the whole space and $\Lambda$ is a $k$-dimensional hyperplane with $k\le \frac{n-2m}{2}$.

arXiv comment34 pages

References in corpus (6)

• Dimensions, Whitney covers, and tubular neighborhoods
• Asymptotic behavior of solutions to the ${\sigma }_k$-Yamabe equation near isolated singularities
• Classification of positive solutions to a nonlinear biharmonic equation with critical exponent
• Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities
• On constant Q-curvature metrics with isolated singularities
• Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities