Some Liouville theorems for the fractional Laplacian

In this paper, we prove the following result. Let a be any real number between 0 and 2. Assume that u is a solution of {(-Delta)alpha/2u(x) = 0, x is an element of R-n, lim vertical bar x vertical bar ->infinity u(x)/vertical bar x vertical bar(gamma) >= 0, for some 0 <= gamma <= 1 and gamma < alpha. Then u must be constant throughout Rn. This is a Liouville Theorem for alpha-harmonic functions under a much weaker condition. For this theorem we have two different proofs by using two different methods: One is a direct approach using potential theory. The other is by Fourier analysis as a corollary of the fact that the only alpha-harmonic functions are affine. (C) 2014 Elsevier Ltd. All rights reserved.