Abstract
In this paper, we prove that if n ≥ 2a ndx0 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 .W e also prove a Bernstein-type theorem, which asserts that if u is a uniformly Lipschitz continuous, one-side bounded infinity harmonic function in R n \{0}, then it must be a cone function with center at 0.
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