NONEXISTENCE AND SYMMETRY OF SOLUTIONS FOR SCHR¨ODINGER SYSTEMS INVOLVING FRACTIONAL LAPLACIAN

. In this paper, we consider the following Schr¨odinger systems involving pseudo-diﬀerential operator in where α and γ are any number between 0 and 2, α does not identically equal to γ .We employ a direct method of moving planes to partial diﬀerential equations (PDEs) (1). Instead of using the Caﬀarelli-Silvestre’s extension method and the method of moving planes in integral forms, we directly apply the method of moving planes to the nonlocal fractional order pseudo-diﬀerential system. We obtained radial symmetry in the critical case and non-existence in the subcritical case for positive solutions. In the proof, combining a new approach and the integral deﬁnition of the fractional Laplacian, we derive the key tools, which are needed in the method of moving planes, such as, narrow region principle, decay at inﬁnity. The new idea may hopefully be applied to many other problems.


1.
Introduction. In recent years, there has been a great deal of interest in using the fractional Laplacian to model diverse physical phenomena, such as anomalous diffusion and quasi-geostrophic flows, turbulence and water waves, molecular dynamics, and relativistic quantum mechanics of stars (see [3], [4], [9], [23]). In particular, the fractional order Laplacian can be understood as the infinitesimal generator of a stable Lévy process (see [2]).

RAN ZHUO AND YAN LI
The fractional Laplacian in R n is a nonlocal pseudo-differential operator, taking the form where α is any real number between 0 and 2 and PV stands for the Cauchy principal value. This operator is well defined in S, the Schwartz space of rapidly decreasing C ∞ functions in R n . In this space, it can also be defined equivalently in terms of the Fourier transform whereû is the Fourier transform of u. One can extend this operator to a wider space of distributions as the following. Let dx < ∞} (see [21]).
For u ∈ L α , we define (−∆) α/2 u as a distribution: Throughout the paper, we consider the solutions in this distributional sense. One can verify that, when u is in S, all the above definitions coincide.
In [24] and [25], the authors considered the same Schrödinger system with high order Laplacian on a upper half space R n + with Navier and Dirichlet boundary conditions: and where R n + is the n-dimensional upper half Euclidean space, R n + = {x = (x 1 , x 2 , · · · , x n ) ∈ R n |x n > 0}, and β 1 , γ 1 , β 2 , and γ 2 satisfy the condition (f 1 ): They considered the corresponding integral systems and where is the Green's function with Navier boundary conditions, and is the Green's function with Dirichlet boundary conditions.
Due to the non-locality of the fractional Laplacian with 0 < α < 2, they technically required that α is any even number between 0 and n in PDEs. They proved that the solutions of the corresponding integral equations must satisfy PDEs. Because of technical limitations, they only conjectured that the converse is also true. By using the method of moving planes in integral forms, they verified Proposition 2. (See Zhuo, Li and Lv [25])For β 1 , γ 1 , β 2 , and γ 2 satisfying (f 1 ), if (u, v) is a pair of non-negative solution of integral systems (7), then u ≡ 0 and v ≡ 0, where α is any real number between 0 and n if n > 3, and 1 < α < n if n = 3. [24]) For β 1 , γ 1 , β 2 , and γ 2 satisfying
Using the equivalence between PDEs and the corresponding integral equations, they partially proved the nonexistence of solutions for PDEs (5) and (6). For more information of this method, please see [1], [12], [13], [15], [16]. In this paper, we directly work on the nonlocal operator to circumvent the difficulty of equivalence between partial differential equations and integral equations.
For fractional Laplacian problems, a useful method is the extension method introduced by Caffarelli and Silverstre (see [6]). By extending the fractional operator to one more dimension, the nonlocal problem becomes a local one taking the form of a second order elliptic equation. Specifically, let u(x) be a function: R n → R, and U (x, y): R n × [0, +∞) → R. Here U (x, y) satisfies the following equation where R n+1 In particular, when α = 1, PDE (9) is reduced to the following problem For (10), one can express the solution U (x, y) by the Poisson kernel P (x, y): here δ 0 (x) is δ-function centered at the origin. Then, However, the extension method can not be applied to uniformly elliptic nonlocal operators and fully nonlinear nonlocal operators. Moreover, it's hard to apply this method to system involving fractional Laplacian. In this paper, we employ the method of moving planes directly to the fractional Laplacian, and derive the symmetry and non-existence of solutions for system involving the fractional operator. For more information of the method of moving planes, please see [10], [11], [8] and [19]. Furthermore, this method can be generalized to study the uniformly elliptic nonlocal problem, such as ( [22]) and hopefully can be applied to equations involving fully nonlinear nonlocal operators.
In the paper, we need some key tools, such as the narrow region principle and decay at in infinity to carry out the method of the moving planes. In Section 2, we will accomplish this. In Section 3, we apply the key technical results in the method of moving planes together with a new idea to show that (3) has no positive solution; (ii) in critical case, that is, (n + α) − (n − α)β 1 − (n − γ)τ 1 = 0 and (n + γ) − (n − α)β 2 − (n − γ)τ 2 = 0, u and v must be radially symmetric with the same center.
In above theorem, we prove the results under the weak condition that u ∈ L α ∩C 1,1 loc , v ∈ L γ ∩C 1,1 loc . Because there is no degeneracy assumption on the solution (u(x), v(x)), we need to apply the Kelvin transform. Let be the Kelvin transform centered at any given point x 0 .
we are able to use the method of moving planes to show thatū andv must be radially symmetric about the point x 0 . Since x 0 is an arbitrary point in R n , we conclude that u, v are constants. This contradicts system (3). This establishes the non-existence of positive solutions.
Based on the proof of above theorem, we will investigate the same system in the half space in our next paper. We firmly believe that we can get similar results in the half space. Together with the results in the whole space and in the half space, we can establish a priori estimates of solutions for a family of nonlocal operators on bounded domains of Euclidean space. In general, let a was introduced in [7] as the uniformly elliptic nonlocal operator, One can use the Liouville type theorems to obtain a priori estimate for solutions of Lu = f (x, u) with the corresponding boundary conditions.
2. Key tools in the method of moving planes. In the section, we will show the key ingredients in the method of moving planes, such as narrow region principle and decay at infinity. First we introduce some basic notation needed in the method of moving planes. For a given real number λ, denote and let be the reflection of the point x = (x 1 , x 2 , · · · , x n ) about the plane T λ .
and c i (x) are positive and bounded from below in Ω, i = 1, 2, then for sufficiently small l, we get For an unbounded narrow region Ω, if we suppose the above conclusions also hold.
Proof of Theorem 2.1. If (13) does not hold, by the lower semi-continuity of U (x) and V (x) on Ω, there exist some points x 0 , x 1 ∈ Ω, such that Actually, by (12), one can further deduce that x 0 and x 1 are in the interior of Ω. By the elementary calculation, we derive Set H = {y = (y 1 , y ) ∈ R n |l < y 1 − x 0 1 < 1, |y − (x 0 ) | < 1}, and let ρ = Let t = ρs, it follows from the above inequality tht where ω n−1 is the area of (n − 1)-dimensional unit sphere, and C denotes some constant.
By (14) and (15), we derive Similarly, we get By the first inequality of (12) and (16), one can see that there exists some constant a 1 > 0, such that for sufficiently small l we have Similarly, there exists some constant a 2 > 0 such that for sufficiently small l, we have Combining (18) with (19), it gives that is 1 It's trivial that the inequality does not hold for sufficiently small l. Hence (13) must be true.
Combining (24) with (25), Similarly, we get It follows from the first inequality of (21) and (26) that Combining this with degenerate assumption of b 1 (x), it's easy to see that there exists some constant C 1 > 0 such that Similarly, there exists some constant C 2 > 0 such that From (28) and (29), we get However, for |x 0 | and |x 1 | sufficiently large, the inequality above is not true. Therefore, there exists R 0 > 0 such that This completes the proof.
3. Rotational symmetry of solutions for Schrödinger system. In this section, we give the proof of Theorem 1. Because these is no decay conditions on u and v in infinity, we apply the Kelvin transform. For any z 0 ∈ R n , consider the Kelvin transform centered at z 0 For simplicity of arguments, we will only show the case when z 0 is the origin, while the proof for a general z 0 is entirely similar. Letū be the Kelvin transform of u and v centered at the origin. It's easy to seeū It is well known that where here be the reflection of the point x = (x 1 , x 2 , · · · , x n ) about the plane T λ . From (35) and (36), it's easy to derive
. By the definition of U λ and V λ , we have This implies that U λ and V λ attain negative minimum in the interior of Σ λ . Define The proof consists of two steps.
Step 1. We will show that, for λ sufficiently negative, By an elementary calculation, we derive that, for For the above inequality, applying the Mean Value Theorem, where ξ and η are valued between x λ and x, c 1 (x) = c (40) By (34), it is easy to derive that Similarly, By (34), we have Suppose there exists some point x 0 such that Similar to (16), for |x 0 | > λ, we get Combining this with (40), we deduce By the degeneracy of b 1 (x) at infinity and (45), for sufficiently negative λ, Next we suppose that there is some point x 1 such that Similar to (44), we obtain Combining (42) and (47), From the degeneracy of c 2 (x) at infinity and (48), for sufficiently negative λ, we have c Combining (46) with (49), we derive that c Using the degeneracy of b 2 (x) and c 1 (x) at infinity, we arrive at c

RAN ZHUO AND YAN LI
That is, For sufficiently negative λ, the inequality does not hold. From Theorem 2.2 (Decay at Infinity), for sufficiently negative λ (or |λ| < R 0 in Theorem 2.2), at least one of U λ and V λ is greater than or equal to 0. Without loss of generality, we assume that To prove (50) also holds for V λ , we argue by contradiction.
From previous arguments of (42) and (47), we know that For the above inequality, combining with (49), we derive that This is a contradiction. And we complete step 1.
Step 1 provides a starting point for us to move the plane T λ to the right along x 1 direction as long as inequality (39) holds. Define In the step, we will prove that λ 0 = 0, and Suppose that λ 0 < 0, we will show that the plane T λ can be moved further more. That is, there exists some small > 0, such that for any λ ∈ (λ 0 , λ 0 + ), we have This is a contradiction with the definition of λ 0 . Therefore, we derive that Actually, for λ 0 < 0, Otherwise, at least one of U λ0 (x) and V λ0 (x) is greater than or equal to zero. Without loss generality, we may assume that U λ0 (x) ≥ 0. That is, there exists some pointx such that U λ0 (x) = min It follows that On the other hand, This is a contradiction with (55). Hence (54) holds. From [26], we have the integral expressions of U λ0 and V λ0 . Combining it with the proof of Appendix A in [14], we can show that there exists c 0 > 0 such that, for sufficiently small , Together with the above bounded-away-from-0 result, we derive that for δ > 0, there exists some constant c 0 > 0 such that For , δ |λ 0 |, 0 λ0 ∈ (Σ λ0−δ\{0 λ 0 } ) ∩ B R0 (0). Since U λ and V λ depend on λ continuously, we have By Theorem 2.2(Decay at infinity), we know that if then there exists a large R 0 such that For sufficiently large R 0 , similar to (46), we obtain Therefore, there exists some pointx such that Meanwhile, for U λ atx, similar to (46), we get By (59) and (60), we have Using the degeneracy of c 1 , we know that c 1 (x) is bounded. Notice that b 2 (x)|x| γ is also bounded for |x| > R 0 . Hence for δ sufficiently small, (61) does not hold. This impliesx ∈ B C R0 ∩ Σ λ will not happen. Employing Theorem 2.1 (Narrow region principle), let the narrow region Ω = (Σ λ \ Σ λ0−δ ) ∩ B R0 (0), while U λ and V λ satisfy system (12), we obtain Hence, This completes the proof of (53). Thus we obtain Similarly, we can move the plane from x 1 = +∞ near to the left, and we can show that Therefore we conclude that Since the direction of x 1 -axis is arbitrary, we derive thatū andv are radially symmetric about the origin.
For any point z 0 ∈ R n , applying the Kelvin transform centered at z 0 (see (30),(31)), 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 u(z 1 ) = u(z 2 ) and v(z 1 ) = v(z 2 ). This implies that u and v must be constants. Positive constant solutions do not satisfy system (3). That is, in subcritical case, there is no positive solution for system (3).
We still use the notations introduced 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.
We consider two possible cases.

RAN ZHUO AND YAN LI
This is a contradiction. Hence (68) holds. When U λ0 ≡ 0, by the anti-symmetry of U λ , that is, we derive that Obviously, it must be true thatv Similarly, if V λ0 (x) = 0 somewhere, then we can prove that U λ0 (x) ≡ 0, x ∈ R n .
When U λ0 (x), V λ0 (x) > 0, x ∈ Σ λ0 \ {0 λ0 }, using an entirely similar argument of Step 2 in subcritical case, one can keep moving the plane T λ . That is, there exists some small > 0, such that for any λ ∈ (λ 0 , λ 0 + ), we have This is a contradiction with the definition of λ 0 . Therefore (67) must not be true. We conclude that This implies u and v are symmetric about some point in R n .
In this case, we can move the plane from near x 1 = +∞ to the left, and derive that This proves thatū andv are symmetric about the origin. So are u and v. In any case, u and v are symmetric about some point in R n . This completes the proof of Theorem 1.