Symmetry and Nonexistence of Positive Solutions for a Fractional Laplacion System with Coupled Terms

: In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: where 0 < α, β < 2 , p, q > 0 and max { p, q } ≥ 1 , α + γ > 0 , β + τ > 0 , n ≥ 2. First of all, while in the subcritical case, i.e. n + α + γ − p ( n − α ) − ( q + 1)( n − β ) > 0, n + β + τ − ( p + 1)( n − α ) − q ( n − β ) > 0, we prove the nonexistence of positive solution for the above system in R n . Moreover, though Doubling Lemma to obtain the singularity estimates of the positive solution on bounded domain Ω. In addition, while in the critical case, i.e. n + α + γ − p ( n − α ) − ( q +1)( n − β ) = 0, n + β + τ − ( p +1)( n − α ) − q ( n − β ) = 0, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of Moving planes in R n .

We also assume that n + α + γ − p(n − α) − (q + 1)(n − β) ≥ 0; n + β + τ − (p + 1)(n − α) − q(n − β) ≥ 0; The system (1.1) and the corresponding parabolic problem appear in the study of static Schrödinger theory and Bose Einstein condensate with two components ( [15]). It also can be used to describe competition of biological population. In recent years, the fractional Laplacian has attracted much attention from the mathematical community due to its nonlocality and widespread applications. It can be used to model diverse physical phenomena. For instance, in the diffusion process, the operator was used to derive heat kernel estimates for many symmetric jump-type processes (see [1]) and to study the acoustic wave equation. In astrophysics, it is used to model the dynamics in the Hamiltonian chaos (see [25]). It also has various applications in probability and finance, in which this operator is defined as the generator of -stable Lvy processes that represent random motions, such as the Brownian motion and the Poisson process (see [8]), anomalous diffusion and quasi-geostrophic flows, turbulence and water waves, molecular dynamics, and relativistic quantum mechanics of stars, see ( [3], [5], [24]) and it also plays an important roles in the theory of nonlinear partial differential equations.
The fractional Laplacian in R n is a nonlocal pseudo-differential operator, we assuming the form (1. 2) where C n,α is a normalization constant and P V represents the Cauchy principal value.

Nonexistence and Symmetry
In this subsection, we study the following system x ∈ R n .
We define the solution of (1.3) in the distribution sense. Let Obviously, the integral in (1.3) is well defined for u ∈ L α ∩ C 1,1 loc , v ∈ L β ∩ C 1,1 loc . It is not easy to deal with the problems involving a nonlocal fractional Laplacian operator due to the nonlocal charactrristic of these operators. The first method to handle such problems is known as the extended method, which was introduced by Caffarelli and Silvestre [4]. Namely, a nonlocal problem ivolving the fractioal Laplacian is transformed into a higher dimensional local prolem.
The study of the solution u : R n → R of the equation involving fractional Laplacian needs its extension U : R n × [0, ∞) → R given by .
The other method is the integral equations method, such as method of moving planes in integral forms to study their equivalent corresponding integrals ( [16], [17]). That is, if we choose the integral equations method to study the well-known nonlinear partical differential equation: we need an equaivalent integral form: Recently, Zhang et al. in [28] studied the problem for a nonlinear elliptic system involving fractional Laplacion: where 0 < α < 2, p, q > 0 and max{p, q} ≥ 1, τ ≥ 0, n ≥ 2. They showed that the positive solution of above system are radially symmetric and decreasing about origin by using the method of M oving planes in R n . Moreover, while in the subcritical case p + q + 1 < n+α+2τ n−α , they proved the nonexistence of positive solution for the above system in R n , then though Doubling Lemma to obtained the singularity estimates of the positive solution on bounded domainΩ.
After routine calculations like in [12], assume u, v ∈ L α ∩ C 1,1 loc and satisfy is defined by the Fourier transform whereû andφ are the Fourier transform of u and φ respectively. Zhang et al. also obtained that system (1.3) as defined above is equivalent to integral system Ma et al. in [19] studied the nonexistence of positive solutions for the following fractional Hénon system where 0 < α < 2, 1 ≤ p, q < ∞, a, b ≥ 0, n ≥ 2. Using a direct method of moving planes, they have been proved the non-existence of positive solution in the subcritical case 1 < p < n+α+a n−α , 1 < q < n+α+b n−α . Moreover, they also proved (1.8) equivalence the following integral system under certain suitable conditions.
In [14], Li concerned the following elliptic system where n ≥ 3, p, q > 0 and max{p, q} ≥ 1. They discussed the nonexistence of positive solution in subcritical case and stable solution in supercritical case, the necessary and sufficient conditions of classification in the critical case, and by using the Liouville theorem of (1.10), they estimated boundary blow-up rate, i.e.
where Ω ⊂ R n is a bounded domain. Chen [6] proved that when n ≥ 3 and min{p, q} > 0, if (u, v) solve (1.10), then it also solves the integral system for some positive constants C 1 , C 2 .
In [2], the author dealed with the local and global behaviour of the positive solution of the semilinear elliptic system in R n (n ≥ 3) where σ, p, q ∈ R, and p, q > 0. Their main results are the fact that the solution satisfy Harnack inequality when p + q + 1 < n+2 n−2 in the local estimates. If not, they also given the precise behaviour of the solution.
Li et al. in [12] studied the following weighted system of partial differential where p, q > 1, 0 < α < n and 0 ≤ s, t < α. They first established the equivalence between partial differential system and weighted integral system (1.14) Then, in the critical case of n−s q+1 + n−t p+1 = n−α, they showed that every pair of positive solution (u, v) are radially symmetric about the origin. While in the subcritical case, they proved the nonexistence of positive solution. Remark 1. Inspired by aforementioned work, we can obtain that system (1.3) as defined above is equivalent to integral system The following is our main theorems. If p and q is subcritical, we will show that there actually is no positive solution.

3) has no positive solutions.
If p and q is critical, we will show that the solution is radical symmetric.
3), u and v must be radially symmetric with the some center.
Based on this Lemma, we establish the equivalence between the Liouville theorem of (1.3) and the estimate of boundary blow-up rate for solutions of (1.16), and combining with the nonexistence of the positive solution of (1.3) we can obtain that the following result.
Thoughout the paper, we use C to denote a generic constant whose value may be different from line to line or even in the same line.
The paper is organized as follows: In Section 2, we given some notations and some necessary lemma. In Section 3, we complete the proof of Theorem 1.1 and Theorem 1.2 by the moving plane. In Section 4, we use the Double Lemma to prove Theorem 1.4.

Preliminaries
In this section, we will given some notations and some necessary lemma to proof of Theorem 1.1 and Theorem 1.2.

Notations
In this subsection, we given some notatons, in order to study the symmetry and monotonicity of positive solutions for fractional systems (1.3) by a direct method of moving planes. We are not able to carry the method of moving planes on u and v directly since there is no any decay conditions on u and v.
To overcome this difficulty, we make a Kelvin transform. Denote be the Kelvin transform centered at any given point z 0 . When z 0 is the origin, while the proof for a general z 0 is entirely similar. Namely, be the Kelvin transform of u and v centered at the origin. It is easy to seē Similarly, we have (2.5) Define the moving planes and the region to the left of the plane be the reflection of the point x = (x 1 , · · ·, x n ) about the plane T λ , and Now, assume that (ū,v) solves the fractional system (1.3). We denote to compare the value ofū λ (x) withū(x) andv λ (x) withv(x) respectively. By system (1.3), we have (2.7)

Necessary Lemma
In order to proof of Theorem 1.1 and Theorem 1.2, we will show the key ingredients in the method of moving planes such as narrow region principle and decay at infinity. Lemma 2.1 (Narrow Region Principle [27]) Let Ω ⊆ {x | λ − l < x 1 < λ} be a bounded narrow region in Σ λ for l > 0 small. Assume that u ∈ L α ∩ C 1,1 loc , v ∈ L β ∩ C 1,1 loc are lower semi-continuous on Ω. If b i (x) and c i (x) are positive and bounded from below in Ω, i = 1, 2, in Ω, (2.8) then for sufficiently small l, we get (2.9) Furthermore, if U (x) = 0 and V (x) = 0 at some point in Ω, then For an unbounded narrow region Ω, if we suppose the above conclusions also hold. Lemma 2.2 (Decay at Infinity) Let Ω be an unbounded region in Σ λ . Assume that u ∈ L α ∩ C 1,1 loc , v ∈ L β ∩ C 1,1 loc satisfy the following equations and b i (x) and c i (x) are nonnegative in Ω, i = 1, 2.
Then, there exists a constant R, which depends on b i (x) and c i (x), i = 1, 2, but is independent of U (x) and V (x), such that if then |x 0 | ≤ R, or |x 1 | ≤ R.
Proof. The proofs are analogous to the ones that [27]. For easily to read, we give outline of the proof. By the elementary calculation, we derive (2.14) For fixed λ, when |x 0 | ≥ λ and |x 1 | ≥ λ, it is easy to derive Combining (2.14) with (2.15), we have Similarly, we obtain (2.17) It follows from the first inequality of (2.10) and (2.16) that That is, However, for |x 0 | and |x 1 | sufficiently large, the above inequality is equivalent to which is a contradiction. Therefore, there exists R > 0 such that This completes the proof.
The following Lemma 2.3 and Lemma 2.4 are also crucical for us in [28].
Lemma 2.3. For λ negative large, there exists a constant C ≥ 0 and ǫ > 0 such that whereλ defined as the following (3.12).
3 Proof of Theorem 1.1 and Theorem 1.2

System in subcritical case
In this subsection, we will use the method of moving planes to prove Theorem 1.1, namely, in the subcritical case, we show that (1.3) has no positive solution.
Proof of Theorem 1.1. By the definition of U λ and V λ , we have The proof consists of two steps.
Step 1. we show that when λ sufficiently negative. (3.1) By an elementary calculation, for x ∈ Σ u λ ∩ Σ v λ , we derive For the above inequality, applying the Mean Value Theorem, we obtain where ξ and η are valued between x λ and x, and Therefore, we obtain It is easy to derive that Similarly, we get Suppose there exists some points x 0 such that We claim that In fact, we have where ω n is the area of n dimensional unit sphere and B x 0 (x 1 ) ⊂ R n \Σ λ with x 1 = (3|x 0 | + x 0 1 , x 2 , · · ·, x n ).
By (2.14) and (3.5), we can derive that (3.4). Combining (3.2) with (3.4), we obtain By the degeneracy of b 1 (x) at infinity and (3.6), for sufficiently negative λ, Now, we suppose that there is some point x 1 such that Similar to (3.4), we obtain From the degeneracy of c 2 (x) at infinity and (3.9), for sufficiently negative λ, we have (3.10) Combining (3.7) with (3.10), we deduce Using the degeneracy of b 2 (x) and c 1 (x) at infinity, we have for sufficiently negative λ, the inequality does not hold.
From Lemma 2.2, for sufficiently negative λ, at least one of U λ and V λ are greater than or equal to 0. Without loss of generality, we assume that The following proves that (3.11) to V λ is also true. In fact, if V λ is negative somewhere in Σ λ \{0 λ }, then there must exist somex ∈ Σ λ \{0 λ } such that From previous arguments of (3.3) and (3.8), we have For the above inequality, combining with (3.10), we deduce It is a contradiction, and then we complete Step 1.
Step 2. The Step 1 provides a starting point, from which we can now move the plane T λ to the right as long as (3.1) holds to its limiting position. Let (3.12) We claim thatλ = 0, If not, we suppose thatλ < 0, we have proved that the plane T λ can be moved further right. Namely, there exists some small δ > 0, such that for any λ ∈ (λ,λ+δ), which is a contradiction with the definition ofλ. Therefore, we deducẽ λ = 0.

By Lemma 2.2, we know that if
then there exists a large R 0 such that For sufficiently large R 0 , similar to (3.4), we obtain Therefore, there exists some pointx such that Ifx ∈ B C R 0 ∩ Σ λ , similar to (3.10), we have (3.18) Meanwhile, for U λ atx, similar to (3.7), we have We know that c 1 (x) is bounded, and b 2 (x)|x| β is also bounded for |x| > R 0 . Hence for ǫ, δ sufficiently small, (3.20) does not hold namely,x / ∈ B C R 0 ∩ Σ λ . Combining with Lemma 2.1, let while U λ and V λ satisfy system (2.8), we have Now, we conclude that neither U λ nor V λ has negative minimum in Σ λ \{0 λ }. Therefore, we obtain This completes the proof of (3.15). Therefore we havẽ Similarly, we can move the plane from x 1 = +∞ near to the left, and we can show that Therefore, we deducẽ Since the direction of x 1 -axis is arbitrary, we haveū andv are radially symmetric about the origin. For any point z 0 ∈ R n apply the Kelvin transform centered at z 0 , and by an entirely similar argument, one can show thatū andv are radially symmetric about z 0 .
Let z 1 and z 2 be any two points in R n and we choose the coordinate system so that the midpoint z 0 = z 1 + z 2 2 is the origin. Sinceū andv are radially symmetric about z 0 , we have This implies that u and v must be constants.
But positive constant solutions do not satisfy system (1.3). Namely, in subcritical case, there is no positive solution for system (1.3).
We still use the notations in the subcritical case. The argument is quite similar to, but not entirely the same as that in the subcritical case. Hence we still present some details here. Proof of Theorem 1.2. In critical case, similar to (3.2), we have The remaining proof is the same as that in the subcritical case.
When (3.21) holds, by using an entirely similar argument of Step 2 in subcritical case. One can keep moving the plane T λ , namely, there exists some small δ > 0, such that for any λ ∈ (λ,λ + δ), we have which is a contradiction with the definition ofλ. Therefore (3.21) must not be true.
We conclude that This implies u and v are symmetric about some point in R n . Case 2.λ = 0 In this case, we can move the plane from near x 1 = +∞ to the left, and derive that U 0 (x), V 0 (x) > 0, ∀x ∈ Σ 0 .