Some Liouville theorems for the fractional Laplacian

Abstract

In this paper, we prove the following result. Let α be any real number between 0 and 2. Assume that u is a solution of {(-δ)α/2u(x)=0,x∈Rn,lim¯|x|→∞u(x)|x|γ≤0, for some 0≥≥;1 and γα. Then u must be constant throughoutRn. This is a Liouville Theorem for α-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 α-harmonic functions are affine

Harnack‐type inequality for fractional elliptic equations with critical exponent

“…and  3R (x) is the ball in R N+1 with radius 3R and its center at the x;  + 3R =  3R ∩ R N+1 + is the upper half ball; and ′  + 3R is the flat part of  + 3R , which is the ball B 3R in R N . For other results of fractional Laplacian equations, please see some works [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and reference therein.…”

Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian

“…We apply Lemma 2 to yield (28), because † is a narrow region for > 1 and sufficiently close to 1. Define…”

Liouville Theorems for a General Class of Nonlocal Operators

“…[7,Theorem 6.2]). In the special case L ν = (−∆) s , this observation has already been used in [4]. At first glance, it is natural to expect that, for a distributional solution u ∈ L 1 s (R N ) of L ν u = 0, the support of u should be disjoint from the largest open set O ⊂ R N where the symbol of Lν does not vanish.…”

“…Very recently, it has been shown by the first author in [9] and independently in [4,Theorem 1.3] that s-harmonic functions in R N are affine if s ∈ ( 1 2 , 1) and constant if s ∈ (0, 1 2 ]. This result generalizes earlier classification theorems stating that bounded or semibounded sharmonic functions are constant, see e.g.…”

“…[2,3,10,15]. The proof in [9] relies on a Poisson kernel representation of s-harmonic functions, whereas the proof of [4,Theorem 1.3] uses Fourier analysis. We point out that the case s = 1 corresponds to a classical theorem of Joseph Liouville stating that bounded harmonic functions on R N are constant.…”