Some Liouville theorems for the fractional Laplacian

In this paper, we prove the following result. Let αα be any real number between 00 and 22. Assume that uu is a solution of {(−△)α/2u(x)=0,x∈Rn,lim¯∣x∣→∞u(x)∣x∣γ≥0, for some 0≤γ≤10≤γ≤1 and γ<αγ<α. Then uu must be constant throughout RnRn.This is a Liouville Theorem for αα-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 αα-harmonic functions are affine.