Non-existence for a semi-linear fractional system with Sobolev exponents via direct method of moving spheres

The aim of this article is to consider the semi-linear fractional system with Sobolev exponents q=n+αn−β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q = \frac{{n + \alpha}}{{n - \beta}}$\end{document} and p=n+βn−α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p = \frac{{n + \beta}}{{n - \alpha}}$\end{document} (α≠β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \ne\beta$\end{document}): {(−Δ)α/2u(x)=k(x)vq(x)+f(v(x)),(−Δ)β/2v(x)=j(x)up(x)+g(u(x)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left \{ \textstyle\begin{array}{l} {( - \Delta)^{\alpha/2}}u(x) = k(x){v^{q}}(x) + f(v(x)),\\ {( - \Delta)^{\beta/2}}v(x) = j(x){u^{p}}(x) + g(u(x)), \end{array}\displaystyle \right . $$\end{document} where 0<α,β<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0 < \alpha, \beta < 2$\end{document}. We first establish two maximum principles for narrow regions in the ball and out of the ball by the iteration technique, respectively. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions to the above system in the whole space and bounded star-shaped domain. As a consequence, the monotonic decreasing properties of W(x)=|x|n−α2u(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W(x) = { \vert x \vert ^{\frac{{n - \alpha}}{2}}}u(x)$\end{document} and W1(x)=|x|n−β2v(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${W_{1}}(x) = { \vert x \vert ^{\frac{{n - \beta}}{2}}}v(x)$\end{document} along the radial direction in the whole space are obtained.


Introduction
In recent years, the fractional Laplacian has been frequently used in the simulation of various physical phenomena, such as anomalous diffusions and quasi-geotropic flows, turbulence and water waves, molecular dynamics, and relativistic quantum mechanics (see [1][2][3][4]). Typically, the standard linear evolution equation involving fractional Laplacian is ∂u ∂t which is a model of anomalous diffusion and has been widely used in physics, probability, and finance [5][6][7]. The fractional Laplacian can be also used to handle Lévy flight, such as the so-called SIS flow progress [8,9], to derive the well-known Kolmogorov-Fisher equation. At present, the study on the fractional Laplace equations has made many important achievements, including the blow up of positive solutions, Liouville theorems, a priori estimates, and so on [10,11].
The fractional Laplacian in R n is a nonlocal pseudo-differential operator, which is of the form (-) α 2 u(x) = C n,α P.V . R n u(x)u(y) |x -y| n+α dy, where 0 < α < 2 and P.V . stands for the Cauchy principal value. Letting it is easy to see that, for u ∈ L α ∩ C 1,1 loc , the integral in the expression of (-) α 2 u(x) is well defined.
Different from the integer order Laplacian, the fractional Laplacian has the so-called nonlocality, which has brought new difficulties in the treatment to equations involving this kind of operators. To circumvent these difficulties, Caffarelli and Silvestre [12] introduced the extension method that reduced the nonlocal problem into a local one in higher dimensions. This method has been extensively applied in the fractional Laplace equations, and a series of results have been obtained (see [13,14]). But by using this method one needs to require 1 ≤ α < 2. One can also consider the method of moving plane in integral forms to investigate the fractional Laplace equations, and this method has been proved to be a powerful tool (see [15][16][17][18][19][20]). However, for some equations including fully nonlinear nonlocal operators, neither the extension method nor the method of moving plane in integral forms can be applied [21,22], and this inspires one to use the direct method of moving plane to inspect these equations. Lately, some success on the direct approach was achieved by Jorohs and Weth in [23], where they raised antisymmetric maximum principles and applied them to show the symmetry of solutions to nonlocal problems by using the direct method of moving planes. It needs to note that their maximum principles are only suitable for bounded regions. Chen, Li, and Li [24] have developed a systematic method of moving plane to nonlocal problems either on bounded or unbounded domains. At the same time, Chen, Li, and Zhang [25], Li and Zhu [26] also used the direct method of moving spheres to the fractional Laplace equations, which is sometimes more convenient than the moving planes in some problems. It is noted that in the method of moving planes, one moves parallel planes along a chosen direction to the limiting position to derive symmetry of the solutions about the limiting plane, while in the method of moving spheres, one fixes a center and increases or decreases the radius of the spheres to conclude monotonicity or symmetry of the solutions along the radial directions of the spheres.
In this paper, we use the direct method of moving spheres to establish non-existence of positive solutions to the following semi-linear fractional Laplace system with Sobolev exponents q and p: in the whole space R n and the bounded star-shaped domain in R n , where Recall that an open set Ω is called star-shaped with respect to origin provided that, for each x ∈ Ω, the line segment {λx|0 ≤ λ ≤ 1} lies in Ω. The elliptic equation with Sobolev exponent n+2 has been extensively researched [27][28][29][30][31][32][33][34][35][36]. As we all know, the properties of solutions to (1.2) closely rely on the coefficient k(x). Ding-Ni in [37] derived the properties of solutions to (1.2) when k(x) = 1η, k(x) = 1, and k(x) = 1 + η respectively, where η(x) is a rotational symmetric function about the region. Lin in [38] proved non-existence of smooth positive solutions to the semi-linear elliptic equation with Sobolev exponent where it is required that r p is non-increasing about r. The conclusions to semi-linear Laplace equations have been extended to fractional Laplace equations. Chen, Li, and Zhang in [25] discussed the fractional describing curvature equation where p = n+α n-α . Via the direct method of moving plane, the Liouville theorem to the fractional Lane-Emden system where 0 < α < 2, 1 < p, q ≤ n+α n-α , was obtained by Cai and Mei [39]. Using the method of moving plane in integral forms, Dou and Zhou in [40] proved the Liouville theorem of the positive solutions to the following fractional Henon system in R n : In this paper, we apply the direct method of moving sphere to obtain non-existence of (1.1). Before describing the main results, let us state some assumptions that are more general than those in the papers mentioned above.
In R n , we assume that (i) f (v(x)) and g(u(x)) are locally bounded positive functions in x, and for r ≥ 0, f (r) and g(r) are non-decreasing in r; (ii) f (r) r q and g(r) r p are non-increasing in r; (iii) k(x) and j(x) are locally bounded positive functions and non-decreasing along each ray {tξ |t ≥ 0} for any unit vector ξ ∈ R n . Theorem 1.1 Assume that (i), (ii), and (iii) are satisfied. If u ∈ L α ∩ C 1,1 loc , v ∈ L β ∩ C 1,1 loc , then system (1.1) has no positive solutions, unless k(x) and j(x) are constants. Theorem 1.1 has the following consequence.
For the bounded star-shaped domain about origin, we need that (iv) f (v(x)) and g(u(x)) are locally bounded positive functions in x, and for r ≥ 0, f (r) and g(r) are non-increasing in r; (v) f (r) r q and g(r) r p are non-decreasing in r; (vi) k(x) and j(x) are locally bounded positive functions and are non-increasing along each ray {tξ |t ≥ 0} for any unit vector ξ ∈ R n .

Theorem 1.3
Let Ω ⊂ R n (n ≥ 3) be a bounded star-shaped domain about the origin and assume that (iv), (v), and (vi) are satisfied. If u ∈ L α ∩ C 1,1 loc and v ∈ L β ∩ C 1,1 loc , then system (1.1) has no positive solution in Ω, unless k(x) and j(x) are constants.
The paper is organized as follows: In Sect. 2, we offer two maximum principles for narrow regions in the ball and out of the ball, respectively, which play important roles in the proofs of main results. It is worth stressing that the iterating technique in the proofs of maximum principles is employed since (1.1) involves two equations. Section 3 is devoted to proving Theorem 1.1 by the Kelvin transform, the maximum principle for the narrow region in the ball, and the direct method of moving spheres. The process in the proof of Theorem 1.1 leads immediately to Corollary 1.2. In Sect. 4, we prove Theorem 1.3 by the maximum principle for the narrow region out of the ball and the direct method of moving spheres.
Throughout the paper, we denote by c a positive constant depending on n, α, and β, which can be different from line to line.

Two narrow region principles
Denote by B λ (0) (λ > 0) a ball centered at the origin and radius λ, and for u(x) and v(x) satisfying (1.1), let Then and hence, then there exists some sufficiently small δ > 0 such that if Proof For |x| large, there exists a positive constant c such that For 0 < δ λ and x ∈ B δ (0) \ {0}, it is easy to see that |x λ | takes large values and hence Choosing λ sufficiently small, it holds Similarly, we can prove that V λ (x) > 1 and the proof of Lemma 2.2 is completed.
For a bounded narrow region Ω in B λ (0), we have the following.
From the proof process in Lemma 2.2, we have Combining it with the first inequality of (2.1), it follows that cU( x) Using (2.5) and the lower semi-continuity of V (x), there exists some x such that Similar to the process of obtaining (2.4), we have For δ < min{δ 1 , δ 2 } sufficiently small, from the second inequality of (2.1), (2.6), It is a contradiction, and hence U( To prove (2.3), we also apply the contradiction and assume that there exists some η ∈ Ω such that V η = 0.
The proof process for Lemma 2.1 (see [25]) shows Combining it with the second inequality of (2.1), it follows c 2 (η )U(η ) > 0 and U η < 0, since c 2 (η ) < 0. This is a contradiction to (2.2) and then V (x) is identically 0 in B λ (0). Thanks to From the second inequality of (2.1) and c 2 (x) < 0, we know But noting Similar to the process of obtaining (2.8), we attain Similarly, if we first assume that there exists some η ∈ Ω such that U(η ) = 0, then one can also check (2.3) following the previous proof. Now Lemma 2.3 is proved.
For a bounded narrow region Ω out of B λ (0), we have the following. Proof: We note that U(x) = -U λ (x) and V (x) = -V λ (x), since U(x) and V (x) satisfy the anti-symmetry. If U(x) ≥ 0 (x ∈ Ω) in (2.10) does not hold, then from the lower semicontinuity of U(x) on Ω, there exists somex ∈ Ω such that (2.12) By virtue of the third inequality of (2.9), we see further thatx is in the interior of Ω.
Letting ξ = λ 2 z |z| 2 , it gives by a similar way to (2.15) that is not identically 0 in R n \ B λ (0), then using (2.16) to (2.21) implies Using the second inequality of (2.9) together with c 4 (x) < 0, we gain U ξ < 0, but it contradicts (2.10). Hence From the second inequality of (2.9) and c 4 (x) < 0, we know we see Similar to (2.23), we achieve At this moment, (2.11) is proved. Similarly, if one assumes that there exists some point ξ ∈ Ω such that U(ξ ) = 0, we can also get (2.11). Lemma 2.4 is proved. Lemma 2.3 will be used to (1.1) in the whole space R n , while Lemma 2.4 to (1.1) in the bounded star-shaped domain.

Proof of Theorem 1.1
The following result is needed in the proof of Theorem 1.1.

Lemma 3.1 Let
, then for ε > 0 sufficiently small, there is a constant c such that Proof Assuming Because of U λ 0 ∈ C 1,1 loc , there exists small δ > 0 such that The integral solution to equation With the similar way to treating J 1 in the proof of Lemma 2.2, we get It follows n-β j(y)u p (y) + g u(y) dy.
Next, let us prove Theorem 1.1.
Proof of Theorem 1.1 Step 1. For λ > 0, let We will show that, for λ > 0 sufficiently small, that is to say, for λ > 0 sufficiently small, To prove (3.4), we first know from Lemma 2.2 that, for λ > 0 sufficiently small, there exists 0 < δ λ such that In the sequel, we will prove B - Similarly, it implies By assumptions (ii) and (iii), we see that g(r) r p and f (r) r q are non-increasing, k(x) and j(x) are non-decreasing along each ray tξ |t ≥ 0 for any unit vector ξ ∈ R n , so and therefore, If V λ (x) < 0, similar to (3.12), it yields If U λ (x) ≥ 0, by assumption (i), g(r) is non-decreasing, and so (3.14) If U λ (x) ≥ 0, similar to (3.14), we achieve After these preparations, let us consider cases for x ∈ B - In fact, Step 1 provides a starting point for moving spheres.
Step 2. Keep moving the sphere B λ until the limiting scale We point out that First, let us prove the following.
Proof We use the contradiction. In Lemma 3.1, we have proved if U λ 0 (x) = 0 or V λ 0 (x) = 0, then for ε > 0 sufficiently small, By the definition of λ 0 , It follows from (3.22) that there exist points x 0 ∈ B λ 0 (0) \ {0} and y 0 ∈ B λ 0 (0) \ {0} such that hence, To prove (3.25), suppose that (3.25) does not hold, then there exists some point that is to say, u λ 0 (x 1 ) = u(x 1 ). By the proof process of Lemma 2.1, we know Therefore, we have from (3.26) that which is a contradiction to (3.24). Hence, Similarly, we can prove Thus (3.25) is proved. Using (3.25), we can show that, for δ 0 > 0 sufficiently small, it holds that, for any λ ∈ [λ 0 , λ 0 + δ 0 ), It will contradict the definition of λ 0 and Claim 1 will be proved. To verify (3.29), we see from (3.25) that, for ε > 0 sufficiently small, By the continuity of U λ and V λ in λ, it implies Noting computations similar to those in Step 1, Lemma 2.3, and the local boundedness of g and f , we arrive at Combining it with (3.30), we derive (3.29). Hence Claim 1 is true. We continue to prove (3.21). Actually, if λ 0 is finite, then from Claim 1 we get (3.31)

Proof of Theorem 1.3
Proof of Theorem 1.3 Let B R (0) be the smallest ball centered at the origin containing Ω. For any λ ∈ (0, R), we denote the set Because of the star-shapedness of Ω, we know that u λ (x) and v λ (x) obtained by the Kelvintransforms of u(x) and v(x) are well defined on λ .

Lemma 2.4 ensures
. Now (4.1) is proved. This provides a starting point for moving spheres. Next we continue to move the spheres.
Step 2. Keep shrinking the sphere B λ until the limiting scale We will show λ 0 = 0. Before proving (4.2), let us note by the definition of λ 0 that It follows from (2.10) that either Because Ω is a star-shaped domain, there exists some point y 0 ∈ ∂Ω such that U λ 0 y 0 > 0, V λ 0 y 0 > 0, and the continuity of U λ 0 (x), V λ 0 (x) implies (4.3).
To prove (4.2), we use the contradiction and assume λ 0 > 0. It allows us to move continuously the sphere ∂B λ 0 and conclude that, for τ > 0 sufficiently small, But this contradicts the definition of λ 0 and it thus proves (4.2).