Liouville theorems for conformal $Q$-curvature equations

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. Moreover, in the case where $K(x)$ is sign-changing, our result improves upon that of Chen-Li (Comm. Pure Appl. Math. 1995: 657-667) even in the classical setting $\sigma = 1$, by removing the asymptotic assumption on the solution at infinity.