Abstract
A fundamental theorem of Liouville asserts that positive entire harmonic functions in Euclidean spaces must be constant. A remarkable Liouville-type theorem of Caffarelli-Gidas-Spruck states that positive entire solutions of −∆u = u n+2 n−2 , n ≥ 3, are unique modulo Möbius transformations. Far-reaching extensions were established for general fully nonlinear conformally invariant equations through the works of Chang-Gursky-Yang, Li-Li, Li, and Viaclovsky. In this paper, we derive necessary and sufficient conditions for the validity of such Liouville-type theorems. This leads to necessary and sufficient conditions for local gradient estimates of solutions to hold, assuming a one-sided bound on the solutions, for a wide class of fully nonlinear elliptic equations involving Schouten tensors. A pivotal advancement in proving these Liouville-type theorems is our enhanced understanding of solutions to such equations near isolated singularities. In particular, we utilize earlier results of Caffarelli-Li-Nirenberg on lower-and upper-conical singularities. For general conformally invariant fully nonlinear elliptic equations, we prove that a viscosity super-(sub-)solution can be extended across an isolated singularity if and only if it is a lower-(upper-)conical singularity. We also provide necessary and sufficient conditions for lower-(upper-)conical behavior of a function near isolated singularities in terms of its conformal Hessian.