Asymptotic Behavior of Infinity Harmonic Functions Near an Isolated Singularity

In this paper, we prove that if n ≥ 2 and x0 is an isolated singularity of a non-negative infinity harmonic function u, then either x0 is a removable singularity of u or u(x) = u(x0) + c|x − x0| + o(|x − x0|) near x0 for some fixed constant c = 0. In particular, if x0 is nonremovable, then u has a local maximum or a local minimum at x0. We also prove a Bernstein-type theorem, which asserts that if u is a uniformly Lipschitz continuous, one-side bounded infinity harmonic function in Rn\{0}, then it must be a cone function with center at 0.