Liouville theorems for conformal $Q$-curvature equations

TL;DR

This paper proves non-existence results for positive solutions of conformal $Q$-curvature equations involving fractional Laplacians, extending Liouville theorems to a broad class of non-local geometric PDEs.

Contribution

It develops a unified approach to establish Liouville theorems for fractional $Q$-curvature equations across all $\sigma ext{ in } (0, n/2)$, overcoming key analytical challenges.

Findings

Liouville theorems for fractional $Q$-curvature equations.

Unified method applicable to all fractional orders.

Addresses challenges due to lack of ODE tools and classification of solutions.

Abstract

In this paper, we study the non-existence of positive solutions for the following conformal $Q$-curvature equation \begin{equation*} (-\Delta)^\sigma u = K(x) u^{\frac{n+2\sigma}{n-2\sigma}} \quad \text{in } \mathbb{R}^n, \end{equation*} where $ \sigma \in (0, n/2)$ is a real number. When $\sigma=1$, this equation reduces to the well-known scalar curvature equation arising from the prescribed scalar curvature problem. For general $\sigma \in (0, n/2)$, it appears in the study of prescribing $Q$-curvature. We establish Liouville theorems under various assumptions on the $Q$-curvature $K(x)$ by developing a unified approach applicable to all $\sigma \in (0, n/2)$. Our method successfully addresses the challenges posed by the absence of ODE tools in the fractional regime and the lack of a classification of Delaunay-type singular solutions for the general fractional Yamabe equation.