LIOUVILLE’S THEOREM FOR A FRACTIONAL ELLIPTIC SYSTEM

. In this paper, we investigate the following fractional elliptic system where 1 ≤ p,q < ∞ , 0 < α,β < 2, f ( x ) and h ( x ) satisfy suitable conditions. Applying the method of moving planes, we prove monotonicity without any decay assumption at inﬁnity. Furthermore, if α = β , a Liouville theorem is established.


1.
Introduction. In recent years, the fractional Laplace equations have been frequently used to describe many science phenomena, such as the turbulence, water waves, anomalous diffusion and quasi-geostrophic flows (see [7], [16], [17], [33] and the references therein). It also has various applications in probability and finance (see [3], [5]).
The fractional Laplacian (− ) α 2 in R n is a nonlocal pseudo-differential operator with the form where 0 < α < 2 and P V stands for the Cauchy principal value. Let It is easy to verify that for u(x) ∈ C 1,1 loc (R n ) ∩ L α (R n ), the integral in (1) is well defined. In this paper, we consider the fractional Laplacian in this space C 1,1 loc (R n ) ∩ L α (R n ).
In the past few decades, the famous Lane-Emden equation x ∈ R n , u(x) > 0, x ∈ R n (2) has been widely studied by many authors ( see [8], [20], [27] ). To the Lane-Emden where p, q ≥ 0, Serrin and Zou [29] derived existence result and confirmed the Lane-Emden conjecture in R 3 on the assumptions that (u(x), v(x)) is bounded by polynomials at infinity. Brandle [4] studied the properties of the positive solutions to fractional Lane-Emden equation (− ) α 2 u(x) = u p (x), x ∈ R n (4) by using the extension method introduced by Caffarelli and Silvestre [6], which turns the nonlocal problem involving the fractional Laplacian into a local one in higher dimensions, and then applied the method of moving planes to show that nonexistence of solutions in the subcritical case.
Existence and uniqueness of positive viscosity solutions to the fractional Lane-Emden system   with p, q > 0, pq = 1 and 1 p+1 + 1 q+1 > n+α n , were established by Leite and Montenegro [22], where Ω ⊆ R n is a smooth bounded domain. Quaas and Xia [28] also considered (5) with Ω = R n + and proved nonexistence of positive viscosity solutions under the condition 1 < p, q < n+2α n−2α . Liouville theorems are very useful in studying semi-linear elliptic equations and systems. For example, they played an essential role in deriving a priori bounds for solutions in [2], [19] and [20], and were used to obtain uniqueness of solutions in [15], [21] and [24]. Recently, equations involving the fractional Laplacian have aroused general concern ( see [1], [7], [14], [23], [36] and the references therein ). The nonlocal nature of fractional operators brings many new difficulties comparing with the usual Laplacian. 2005, Chen, Li and Ou [12] introduced the method of moving planes in integral forms to the fractional Laplacian equations. Subsequently, Chen, Li and Li [10] developed a direct method of moving planes which can be used directly to the nonlocal operators. These methods have been applied to semi-linear differential equations, free boundary problems and other problems, and a series of fruitful results have been obtained (see [9], [18], [26], [34], [35], and the references therein ).
In this paper, we consider the following fractional elliptic system   where 1 ≤ p, q < ∞, 0 < α, β < 2. Since we do not assume any decay assumption on the solutions at infinity, we construct a function and combine the method of moving planes to derive monotonicity. If α = β, we obtain nonexistence of positive solutions to system (6).
The following is our main result: Theorem 1.1. Assume that 0 < α, β < 2, and 1 ≤ p, q < ∞. Then the positive bounded solutions of system (6) in (C 1,1 loc (R n ) ∩ L α (R n )) × (C 1,1 loc (R n ) ∩ L β (R n )) are monotone increasing along the x 1 direction under the following conditions . The paper is organized as follows. In Section 2 we construct the function g(x) and use the method of moving planes to show that every positive bounded solution to system (6) must be monotone increasing along the x 1 direction. The proof of Theorem 1.2 is given in Section 3.
Throughout the paper, we use C to denote positive constants whose values may vary from line to line.

2.
Monotonicity of positive solutions. In this section, we prove Theorem 1.1.
Proof of Theorem 1.1. We will show that every positive bounded solution to system (6) must be monotone increasing along the x 1 direction.
For λ ∈ R, denote by T λ = {x ∈ R n |x 1 = λ} the moving plane, by Σ λ = {x ∈ R n |x 1 < λ} the left region of the plane T λ , by the reflection of x about T λ , and let Step 1. Move the plane T λ from −∞ to the right along the x 1 -direction.
We claim that for λ sufficiently negative, In fact, it follows by the mean value theorem that where η λ (x) is between v λ (x) and v(x).
To prove (7), we apply a contradiction argument. Without loss of generality, suppose that U λ (x) is negative at some point in Σ λ .
Proof. We only consider that U λ (x) < 0 at some points and another situation is similarly treated. Notice that u(x) and v(x) do not have the decay condition near the infinity and the usual approaches for the case are to use the Kelvin transform or build an auxiliary function. Because system (6) contains f (x) and h(x), it is going to be complicated to use the Kelvin transform. So we construct a function and σ is a small positive number to be chosen later. LetŪ .
(9) An elementary calculus gives with a positive constant C 1 and |S r | being the n − 1 dimensional sphere with radius r.
We split I 2 into three parts: To calculate I 21 , we need a known lemma.
where C t depends continuously on t,

Lemma 2.1 implies
dz by the coordinate transformation. C σ > 0 is as small as we wish for sufficiently small σ.
Next we evaluate I 22 . Consider two regions In where ξ ∈ Σ λ is some point on the line segment fromx to y and C 2 is some positive constant depending only onx. Consequently, On D 2 , since |g(x) − g(y)| ≤ |g(x)| + |g(y)| ≤ C 3 |y| σ and |x − y λ | ≥ |x − y| ∼ |y|, where C 3 is some positive constant depending only onx. The proof of the final inequality in (14) is as follows.
Combining (13) and (14), Hence Taking into account of (10) and (16), we obtain from (9) that where we choose a large K and then a sufficiently small σ (hence C σ becomes sufficiently small), such that C 1 − C σ − C 2 σ − C3 K α−σ is a positive constant, which guarantees I 1 + I 2 > 0.
Now we continue to prove (7). Let Obviously,V λ (x) and V λ (x) have the same sign and lim |x|→∞V λ (x) = 0. So there exists x such thatV Similarly to (9), we conclude where |x−y λ | n+β dy. Since the estimates to J 1 , J 2 are similar to I 1 , I 2 respectively (just replace α with β), it follows By the mean value theorem, we derive where ξ λ (x) is between u λ (x) and u(x). From (8) and (17), it shows Similarly, combining (21), (22) and (23), we have But this contradicts with the boundedness of u(x), v(x) and the condition (i). So we obtain (21) and (22) that (7) is correct.
It implies by (24) that Denote , v k (x)) also satisfies (6). So we also have 1 Noticing that u k (x), v k (x), (− )  2 v k (x) are bounded, by the fractional Sobolev embedding [25] and regularity of solutions to fractional Laplace equations [11], one can derive that {u k (x)} and {v k (x)} are convergent. We will prove that there exist nonnegative functionsũ(x) andṽ(x) ( ≡ 0) such that as k → ∞, and v k (x) →ṽ(x) and (− ) To verify (30) and (31), we need to establish a uniform C 0,α+θ estimate for u k in a neighborhood of any point x ∈ R n , which is independent of k and x. This is done in two steps. We first obtain a C θ estimate ( 0 < θ < 1 ), and then boost C θ up to C 0,α+θ by using the equation satisfied by u k (x). For convenience, we denote We recall three known propositions.
Since u(x) and v(x) are positive bounded solutions to system (6), f k (x) and h k (x) are bounded, we have |u k (x)| ≤ C, |v k (x)| ≤ C, and Hence for any x 0 ∈ R n , B 3 (x 0 ) ⊂ R n , it implies by proposition 3 that and where C is independent of k and x 0 . Let us first consider {v k }. Due to the above uniform estimate (33), it follows from the Arzelà-Ascoli theorem that there exists a converging subsequence of {v k } in B 1 (0), denoted by {v 1m }. Then one can find a subsequence of {v 1m }, denoted by {v 2m }, that converges in B 2 (0), and then a subsequence of {v 2m }, denoted as {v 3m }, converging in B 3 (0). By induction, we get a chain of sub-sequences such that {v jm } converges in B j (0) as m → ∞. Taking the diagonal sequence {v jj }, it sees that {v jj } converges at all points in any B R (0). Thus we have constructed a subsequence (still denoted by {v k }) of solutions that converges point-wisely in R n to a functionṽ(x). From the condition (ii), it follows f k (x) → f (x).
Similarly, we also have u k (x) →ũ(x) and h k (x) → h(x).
Next we show that (− ) β/2 v k (x) also converges point-wisely to (− ) β/2 v(x). To do so, we use the equation and derive from (32) that for any x o ∈ R n and some 0 < θ < 1, where C is independent of k and x o . Now applying proposition 2 to the equation (34), we show that there exists a positive constance C independent of k and x 0 , such that v k C 0,β+θ (B1(x 0 )) ≤ C.
3. Liouville's theorem. In this section, we prove Theorem 1.2. Its proof is based on the following Maximum Principle: Then u(x) ≥ 0, x ∈ R n . If u(x) = 0 at some point in Ω, then u(x) = 0 almost everywhere in R n .
So far, we claim that (42) is not valid. Similarly, one can prove that (43) is not also valid.
Therefore, system (6) possesses no bounded positive solution with α = β. This completes the proof of Theorem 1.2.