Local estimates for conformal $Q$-curvature equations

Abstract

We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$(-\Delta {)}^{m}u=K(x)u^{\frac{n+2m}{n-2m}}in\Omega \backslash \Lambda$$ near their singular set $\Lambda$, where $\Omega \subset {\mathbb{R}}^n$ is an open set, $K(x)$ is a positive continuous function on $\Omega$, $\Lambda$ is a closed subset of ${\mathbb{R}}^n$, $2\le m<n/2$ and $m$ is an integer. Under certain flatness conditions at critical points of $K$ on $\Lambda$, we prove that $u(x)\le C[dist(x,\Lambda ){]}^{-(n-2m)/2}$ when the upper Minkowski dimension of $\Lambda$ is less than $(n-2m)/2$.

57 pages

References in corpus (3)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• On constant Q-curvature metrics with isolated singularities
• Isolated singularities of solutions to the Yamabe equation in dimension $6$