A Liouville Theorem for Möbius Invariant Equations

In this paper, we classify Möbius invariant differential operators of second order in two-dimensional Euclidean space, and establish a Liouville type theorem for general Möbius invariant elliptic equations. The equations are naturally associated with a continuous family of convex cones ${\mathrm{\Gamma }}_p$ in ${\mathbb{R}}^2$, with parameter $p\in [1,2]$, joining the half plane ${\mathrm{\Gamma }}_1:=\{({\lambda }_1,{\lambda }_2):{\lambda }_1+{\lambda }_2>0\}$ and the first quadrant ${\mathrm{\Gamma }}_2:=\{({\lambda }_1,{\lambda }_2):{\lambda }_1,{\lambda }_2>0\}$. Chen and C. M. Li established in 1991 a Liouville type theorem corresponding to ${\mathrm{\Gamma }}_1$ under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville type theorem we establish in this paper for ${\mathrm{\Gamma }}_p$, $1<p\le 2$, does not require any additional assumption on the solution as for ${\mathrm{\Gamma }}_1$. This is reminiscent of the Liouville type theorems in dimensions $n\ge 3$ established by Caffarelli, Gidas and Spruck in 1989 and by A. B. Li and Y. Y. Li in 2003–2005, where no additional assumption was needed either. On the other hand, there is a striking new phenomena in dimension $n=2$ that ${\mathrm{\Gamma }}_p$ for $p=1$ is a sharp dividing line for such uniqueness result to hold without any further assumption on the solution. In dimensions $n\ge 3$, there is no such dividing line.