Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian

In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}u+au=f(v), \\ (-\Delta)^{\frac{\beta}{2}}v+bv=g(u), \end{cases}\displaystyle \quad \text{on } \Omega\subseteqq \mathbb{R}^{n}, $$\end{document}{(−Δ)α2u+au=f(v),(−Δ)β2v+bv=g(u),on Ω⫅Rn, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a,b\geq0$\end{document}a,b≥0. When Ω is the unit ball or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document}Rn, we prove that the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(u,v)$\end{document}(u,v) are radially symmetric and decreasing. When Ω is the parabolic domain on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document}Rn, we prove that the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(u,v)$\end{document}(u,v) are increasing. Furthermore, if Ω is the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}_{+}$\end{document}R+n, then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.


Introduction
This paper is mainly devoted to investigating the properties of the solutions of the following system involving the fractional Laplacian operators:  where P. V. stands for the Cauchy principle value, C n,α , C n,β > 0 and 0 < α, β < 2. To make sense for the integrals, we require u ∈ C 1,1 loc ∩ L α , v ∈ C 1,1 loc ∩ L β , where For more background on the fractional Laplacian operator (-) α 2 , we refer to [1][2][3][4]. We mention that there are also several applications involving the fractional Laplacian in mathematical physics [5][6][7][8], finance [9], image processing [10], and so on.
Since the fractional Laplacian is nonlocal, that is, it does not act by pointwise differentiation but as a global integral with respect to a singular kernel, this is the main difficulty in studying problems involving it. To circumvent this difficulty, Caffarelli and Silvestre [11] introduced the extension method (CS extension) to overcome the difficulty of nonlocality. Their idea is to localize the fractional Laplacian by constructing a Dirichlet to Neumann operator of a degenerate elliptic equation. We can also use the integral equation method, the method of moving planes in integral forms, and regularity lifting to investigate equations involving the fractional Laplacian. Recently, Chen, Li, and Li [12] developed a new method that can handle directly these nonlocal operators. They used this property to develop some techniques needed in the direct method of moving planes in the whole space R n and the upper half-space R n + , such as the narrow region principle and decay at infinity. The direct method of moving planes is very useful, and a series of fruitful results have been obtained. For more articles concerning the method of moving planes for nonlocal equations and systems, mainly for integral equations, we refer to [13][14][15][16][17][18][19][20][21].
In this paper, following the ideas of [12], among others, we consider the properties of the solutions to system (1.1) for different domains . More precisely, we get the following four theorems. Firstly, we consider the case where is the unit ball. For simplicity, we denote B = B 1 (0). We have with M > f (·), g (·) > 0, where M is a positive constant. Then u is radially symmetric and decreasing about the origin.
Remark 1. 1 Li [22] considered the similar problem with f (v) = v p and g(u) = u p . So Theorem 1.1 can be regarded as an extension of the result in [22].
We denote by the parabolic domain on R n .
with M > f (·), g (·) > 0, where M is a positive constant. Then u, v are increasing in x n . Now we consider the whole space case.
for some a, b ≥ 0. (1.4) Suppose that, for γ , ν > 0, and 0 < f (s) ≤ s p and 0 < g (s) ≤ s q with pγ ≥ α and qν ≥ β. Theorem 1.4 Let u ∈ L α (R n + )∩C 1,1 loc (R n + ) and v ∈ L β (R n + )∩C 1,1 loc (R n + ) be nonnegative solutions of the system for some a, b ≥ 0. Remark 1.2 In Sect. 2, we introduce two maximum principles, namely, the narrow region principle and decay at infinity. This two maximum principles play a key role in the proof of Theorems 1.1-1.4. We give detailed proofs of our main theorems in Sect. 3.

Two maximum principles
Let T λ be a hyperplane in R n . Without loss of generality, we assume that be the reflection of x about the plane T λ . Set For simplicity of notations, we denote U λ (x) by U(x) and V λ (x) by V (x).

Lemma 2.1 (Narrow region principle) Let be a bounded narrow region in λ that is contained in
loc ( ) and suppose that U, V are lower semicontinuous on . Assume that C 1 (x) and C 4 (x) are bounded from below in , whereas C 2 (x), C 3 (x) < 0 are bounded from below in . If U, V satisfy the system then, for sufficiently small l, we have

These conclusions hold for an unbounded domain if we further assume that
Remark 2.1 After finish this paper, we have found that this lemma was given by Niu and Wang [23]. For completeness, we give a full proof of the lemma with some changes.
Proof of Lemma 2.1 Suppose on the contrary, that (2.2) is false. Without loss of generality, we assume that there exists a point such that Then by the definition of (-) (2.5) To estimate I 1 , we notice that (2.6) Since U(y) ≥ 0 and y ∈ λ \ , by (2.4) we have We can see that Combining the previous estimate with (2.7), we have which is equivalent to Since we can choose l small enough and C 1 is bounded from below, we have C by the condition C 2 (x) < 0 and (2.4) we get that On the other hand, V is lower semicontinuous on ; hence there exists x 0 such that Similarly to (2.8), we derive that Combining (2.8) and (2.12) with (2.1), we have .
We notice that C 4 (x) is bounded from below in and C 2 (x), C 3 (x) < 0 are bounded from below in . Choosing l small enough, we can derive that V (x 0 ) ≥ 0. This yields a contradiction with (2.11). So (2.2) holds.
Furthermore, if is an unbounded domain, then by the decay condition of U, V it is easy to see that the negative minimum of U, V cannot be taken at infinity. Now we prove (2.3). Without loss of generality, we assume that there exists x 0 ∈ such that U(x 0 ) = 0. Then, due to (2.2) and the fact that C 2 (x 0 )V (x 0 ) ≤ 0, combining (2.5) with the first equation of (2.1), we have If U(y) ≡ 0, y ∈ λ , then noticing that U(y) ≥ 0, y ∈ λ , we have This yields a contradiction. So By (2.10) we immediately get V (x) ≡ 0, x ∈ λ . So since U and V are antisymmetric functions, we have (2.3).

13)
where C 1 (x), C 4 (x) are nonnegative on , whereas C 2 (x), C 3 (x) < 0 on , and furthermore, Then there exists a constant R 0 > 0 (depending on C i (x) but independent of U, V ) such that if Remark 2.2 This lemma is quite the same as that of Niu and Wang [23] but with some difference. In [23], C 1 , C 4 should satisfying So we conclude that the condition of our lemma is different with [23], even they both have the same result.
Proof of Lemma 2.2 Without loss of generality, we assume that there exist a point x 0 ∈ such that Then as in the proof as (2.7), we have Choose a point in c λ : x * = (3| x| + x 1 , x ), where x = (x 1 , x ). Then B | x| ( x * ) ⊂ c λ , and there exists C > 0 such that Combining this with (2.18), we get that Now by the conditions C 1 ( x) ≥ 0 and C 2 ( x) < 0 we easily calculate that Noticing that U( x) < 0 and C 2 ( x) < 0, we get V ( x) < 0. Then since V are lower semicontinuous on . there exists x 0 such that (2.20) Similarly to (2.18), we derive that Choosing | x|, |x| large enough, by (2.14) we can derive V (x) ≥ 0. This yields a contradiction with (2.20). So the lemma is proved.

Proof of Theorems 1.1-1.4
Proof of Theorem 1.1 Let T λ , x λ , u λ , λ , and U λ , V λ be defined as in the previous section.
Step 1: We will show that for λ > -1 and sufficiently close to -1, we have By the first equation in system (1.2) and the mean value theorem it is easy to see that where ξ (x) is between v(x) and v λ (x). Similarly, we also have where η(x) is between u(x) and u λ (x). Choosing C 1 = a, C 2 = -f (ξ (x)), C 3 = b, and C 4 = -g (η(x)), by the narrow region principle (Lemma 2.1) we get (3.1).
Step 2: Define Then we claim that Suppose the claim is not true. If λ 0 < 0, then we will show that the plane can be moved to the right a little more so that inequality (3.1) will still valid. More precisely, there exists small > 0 such that, for all λ ∈ [λ 0 , λ 0 + ), inequality (3.1) holds, which contradicts the definition of λ 0 .
First, since U λ 0 and V λ 0 are not identically zero, from the proof of the narrow region principle (Lemma 2.1) we have Thus U λ 0 and V λ 0 can take the minimum values if x ∈ λ 0 -δ ∩ B. More precisely, for any δ > 0, By the continuity of U λ and V λ with respect to λ there exists > 0 such that We can see that ( λ \ λ 0 -δ ) ∩ B is a narrow region if and δ are small enough. Then by the narrow region principle (Lemma 2.1) we have This contradicts the definition of λ 0 . Therefore we prove the claim (3.3). It follows that or, more apparently, Since the x 1 -direction can be chosen arbitrarily, it follows that u, v are radially symmetric about the origin. The monotonicity is a consequence of the fact that for all -1 < λ ≤ 0. This completes the proof of Theorem 1.1.
Proof of Theorem 1.2 Let T λ , x λ , u λ , and U λ , V λ be defined as in the previous section. Let Step 1: Similarly to (3.1), we can get that, for λ > 0 sufficiently close to 0, we have Step 2: Define Then we must have Otherwise, suppose that λ 0 < +∞, Then we claim that If (3.5) were not true, then by the narrow region principle (Lemma 2.1) we would have We will show that the plane T λ can be moved further to the right. More precisely, there exists small > 0 such that, for all λ ∈ [λ 0 , λ 0 + ), This is a contraction with the definition of λ 0 . If (3.6) holds, then for any δ > 0, By the continuity of U λ , V λ with respect to λ there exists > 0 such that We can see that λ \ λ 0 -δ is a narrow region since and δ are small enough. Then by the narrow region principle (Lemma 2.1) we have This contradicts the definition of λ 0 . Then the claim (3.5) holds, which implies which contradicts the fact that u, v > 0 on . We have shown that λ 0 = ∞ and U ≥ 0, V ≥ 0. This shows that u(x) and v(x) are increasing in x n , which completes the proof of Theorem 1.2.
Proof of Theorem 1.3 Start moving the plane T λ from -∞ to that right along the x 1direction.
Step 1: We will show that for λ sufficiently negative, By the first equation in system (1.4) and the mean value theorem it is easy to see that where ξ (x) is between v(x) and v λ (x). Similarly, we also have where η(x) is between u(x) and u λ (x). In fact, At those points, for |x| sufficiently large, the decay assumptions (1.5) and (1.6) immediately yield and going through a similar proof, we have lim |x|→∞ g η(x) |x| β = 0.
Then by the decay at infinity (Lemma 2.2) there exists a constant R 0 > 0 such that for λ < -R 0 in Lemma 2.2, one of U λ (x) and V λ (x) must be positive in λ . Without loss of generality, we assume that Now we can also prove U λ (x) ≥ 0, x ∈ λ . If not, then by the decay condition of u(x) there must exist a point x 0 ∈ λ such that (3.10) From previous arguments (2.18) and (3.9) we have and then We can derive that U λ (x 0 ) ≥ 0, which contradicts with (3.10). So (3.8) holds.
Step 2: Keep moving the planes to the right to the limiting positive T λ 0 as long as (3.8) holds. Let We have that Otherwise, if λ 0 = ∞, then the solution u(x) is increasing with respect to x 1 . This contradicts condition (1.5), so that λ 0 < ∞. Then we claim that If (3.11) were not true, then by the proof of the narrow region principle (Lemma 2.1) we would have We will show that the plane T λ can be moved further to the right. More precisely, there exists small > 0 such that, for all λ ∈ [λ 0 , λ 0 + ), This is a contraction with the definition of λ 0 .
If (3.12) is true, then let R 0 be determined in the decay at infinity (Lemma 2.2). It follows that, for any δ > 0, Since we have the continuity of U λ (x) and V λ (x) with respect to λ, there exists > 0 such that, for any λ ∈ [λ 0 , λ + ), Suppose (3.13) is not true. If x 0 and x 0 are the negative minima of U λ (x) and V λ (x) in λ , then by decay at infinity (Lemma 2.2) and (3.14) we can get that they are all in the bounded narrow region ( λ 0 + \ λ 0 -δ ) ∩ B R 0 (0) for δ and small enough, which contradicts the narrow region principle (Lemma 2.1). So (3.13) has to be true, which is a contradiction with the definition of λ 0 . Now we have proved the claim (3.11). Since the x 1 -direction can be chosen arbitrarily, we get that u(x) and v(x) are radially symmetric and decreasing about some point x 0 . This completes the proof of Theorem 1.3.
Proof of Theorem 1.4 First, we claim that To prove (3.15), we assume that u(x) ≡ 0. If there exists x 0 ∈ R n + such that u(x 0 ) = 0, then we have that (-) α 2 u(x 0 ) = C n,α P. V. On the other hand, by the function (1.7) and the condition on f (x) and g(x) we have that (-) This yields a contradiction, so we have that either u ≡ 0 or u > 0 in R n + .
Step 1. For λ > 0 sufficiently close to 0, set = λ ∪ R n -, where R n -= {x ∈ R n | x n ≤ 0}. First, we know that U(x) = u λ (x)u(x) ≥ -u(x) and V (x) = v λ (x)v(x) ≥ -v(x), so by condition (1.8) we have Then by the narrow region principle (Lemma 2.1) it is easy to see that since λ is a narrow region.
Therefore we have proved the claim λ 0 = ∞, and consequently the solutions u(x) and v(x) are increasing with respect to x n . We recall that condition (1.8) tells us that lim |x|→∞ u(x) = 0 and lim |x|→∞ v(x) = 0. So the claim (3.16) is not true, and thus u(x) ≡ 0 and v(x) ≡ 0, x ∈ R n .