A priori bounds and existence result of positive solutions for fractional Laplacian systems

In this paper, we consider the fractional Laplacian system \begin{document}$\left\{\begin{array}{ll}(-\triangle)^{\frac{\alpha}{2}}u+\sum^{N}_{i = 1}b_{i}(x)\frac{\partial u}{\partial x_{i}}+C(x)u = f(x,v), \;\;x\in \Omega,\\(-\triangle)^{\frac{\beta}{2}}v+\sum^{N}_{i = 1}c_{i}(x)\frac{\partial v}{\partial x_{i}}+D(x)v = g(x,u),\;\; x\in \Omega,\\u>0, v>0, \;\; x\in \Omega,\\u = 0, v = 0, \;\; x\in \mathbb R^{N}\setminus \Omega,\end{array}\right.$ \end{document} where \begin{document}$Ω$\end{document} is a smooth bounded domain in \begin{document}$\mathbb R^{N}$\end{document} , \begin{document}$α ∈ (1,2)$\end{document} , \begin{document}$β ∈ (1,2)$\end{document} , \begin{document}$N>\max\{α, β\}$\end{document} . Under some suitable conditions on potential functions and nonlinear terms, we use scaling method to obtain a priori bounds of positive solutions for the fractional Laplacian system with distinct fractional Laplacians.

1. Introduction. We consider the following fractional Laplacian system where Ω is a smooth bounded domain in R N , α ∈ (1, 2), β ∈ (1, 2), N > max{α, β}. Under some suitable conditions on potential functions and nonlinear terms, we want to obtain a priori bounds of positive solutions for the fractional Laplacian system.
The fractional Laplacian (− ) α 2 u(x) is defined by where P.V. denotes the Cauchy principal value of the integral ( see [15] ). The fractional Laplacian appears in many areas, for example, physics, probability, mathematical finances and so on, see [2,6,14] and reference therein. The fractional Laplacian has been extensively studied during the last decade in mathematics. When we consider the scalar fractional Laplacian equations, a large number of articles have used variational methods to obtain existence and multiplicity of solutions for the fractional Laplacian equations, see [1,3,4,12,13,16,17,19,20,26,31,32,33,34,35] and reference therein. Recently, Chen (2) when pq = 1, p, q > 0 and By a monotonicity result, Quaas and Xia in [28] obtained the non-existence of positive viscosity solutions for the following system   where 1 < p, q < N +α N −α . Recently, Li and Ma in [25] used the iteration method and a direct method of moving planes to consider the the following system where f ≥ 0 and g ≥ 0 are non-decreasing continuous and they satisfy some more conditions.
When the fractional Laplacian equations and systems are not of variational type, we can not use variational methods to solve the problems. As we know, the related results are considered in few articles.
For the fractional Laplacian equation and (− ) by the classical scaling method of Gidas and Spruck and topological degree, Barrios et al. in [5] obtained a priori bounds and existence of positive solutions for equation (5) and equation (6) under the condition 1 < p < N +α N −α . Recently, Leite and Montenegro in [23] considered the following strongly coupled fractional Laplacian system in non-variational form They established a priori bounds of positive solutions for subcritical and superlinear nonlinearities by means of blow-up method. They also derived the existence of positive solutions through topological method.
It is worth to point out that our paper is inspired by [30]. For the fractional Laplacian system where 1 < α < 2, β 11 > 1, β 22 > 1, and some suitable conditions hold, Quaas and Xia in [30] obtained a priori bounds and existence of positive solutions for system (8). It is a natural question: do we have a priori bounds if the system contains linear terms u, v, ∇u and ∇v? Furthermore, as far as we know, there are few works about the fractional system with distinct fractional operators. In our paper, we solve the problem when the fractional operators are distinct.
To state our main result with respect to system (1), let us introduce the notations (cf. Chapter 6 in [18]) and where γ ∈ R, u ∈ C 1 (Ω) and d(x) := dist(x, ∂Ω). We now formulate the assumptions.
is a positive viscosity solution of system (1), then there exists a positive number C, such that u 1,θ ≤ C, v 1,θ ≤ C.
When the potential functions b i (x) = c i (x) = C(x) = D(x) = 0, ∀x ∈ Ω, then system (1) turns into the following system Similar to the proof of Theorem 3.1 in [30], we can have a priori bounds for system (10).
is a positive viscosity solution of system (10), then there exists a positive number C, Furthermore, using topological degree and the priori bounds in Theorem 1.2, we can prove the existence of positive solutions for system (10).
This paper is organized as follows. In Section 2, preliminary results are revisited. We prove Theorem 1.1 in Section 3. In Section 4, we obtain some results and prove Theorem 1.3.

Preliminaries.
To begin with, we introduce some notations. For x ∈ Ω, d(x) = dist(x, ∂Ω). For r > 0, B r := {x ∈ R N : |x| < r} represents the ball with radius r. We shall use C and C i to represent positive constants which may be distinct even in the same line.
then there exists a positive constant C such that where β 1 = α+qβ pq−1 , β 2 = β+pα pq−1 . Proof. Assume on the contrary that there exists a sequence {(u k , v k )} of positive solutions of system (12) and By direct calculation, we know that ( u k , v k ) is a positive solution of the following system

LISHAN LIN
For k large enough, we obtain that and So u k , ∇ u k , v k , ∇ v k are uniformly bounded in B k . By Theorem 2.2, we get that there exist β ∈ (0, 1), β ∈ (0, 1), such that u k ∈ C 1,β loc (Ω), v k ∈ C 1,β loc (Ω). For any B R ⊂⊂ Ω, there exists a constant C = C(R) > 0, such that By Ascoli-Arzelá , s theorem and the diagonal argument, up to a subsequence, we have that u k → u, v k → v in C 1 loc (R N ) as k → ∞. Taking limit in (15), we have By the fact that β + β 2 > 0 and (F 1 ), we have If the case (I+ III) holds, then we may assume x k → x 0 . Taking the limit in system (13) by Theorem 2.3, we have But by Theorem 1.2 in [23] we know that system (21) has no positive solution. So We get the contradiction.
If the case (I+ IV) holds, then we may assume x k → x 0 . Taking the limit in system (13) by Theorem 2.3, we have Using Liouville theorem to the second equation in system (22), we obtain v ≡ C ≥ 0. But by the case (IV), We can know u k (x) → 0. Therefore we get u ≡ 0. By the first equation in system (22), we have v ≡ 0. So we get the contradiction.
In the same way, we can also get the contradiction in case (II+ III) and in case (II+ III). Thus we get the result of Lemma 2.6.
3. Result about a priori bounds. We prove Theorem 1.1 in this section.
Proof of Theorem 1.1. Assume on the contrary that there exists a sequence {(u k , v k )} of positive solutions of system (1) such that u k 1,θ → ∞ or v k 1,θ → ∞, as k → ∞.
We may assume that here β 1 , β 2 are positive numbers which are the same as the numbers in Lemma 2.6. Denote There exists By direct calculation, we can deduce that (u k , v k ) is a positive solution of the following system Now we want to estimate u k 1,θ .
It is easy to see that For x ∈ Ω k , by direct calculation and (23), we have and Choosing the point z k = x k −ξ k λ k , by the definition of M k and choice of x k , we know that There exists a subsequence of {x k } which is also denoted by {x k } such that x k → x 0 ∈Ω. By virtue of Lemma 2.6 and the boundedness of d(x k ) , we obtain By (28), we have Combining the definition of λ k with (29), we obtain that Here we require pq > 1, θ < qβ+α pq−1 , so β1 β1−θ > 0. Up to a subsequence of {x k }, we can get that If there exists a subsequence of {v k } ( we still denote it by {v k }) such that λ −β2 We can use (25) to obtain and Combining (24), (32), (33) with (34), we deduce that Because of (35), we use Lemma 2.4 to deduce that By (25), we have Therefore We take δ > C . By Lemma 5 in [5], we get that |∇u k | ≤ Cd −θ−2+α k (x). So we can get that We can use (27) together with (37) to deduce that d k (z k ) is bounded away from 0 as k large enough. Using the fact d k (z k ) = d(x k ) λ k , we know d > 0. By the fact 0 ∈ ∂Ω k , we can obtain that z k is also bounded away from 0. Thus up to a subsequence, By (F 2 ) and Theorem 1.3 in [23], system (38) has no positive solution. Thus we deduce the contradiction. Therefore we get the result of Theorem 1.1.

4.
Proof of Theorem 1.3. In this section, we use the following important topological degree theory to consider the existence of positive solutions for system (10). Given (u, v) ∈ C(Ω) × C(Ω), we consider the following system Then system (39) has a unique nonnegative solution (ū,v) with |ū| L ∞ < C, |v| L ∞ < C. We define We consider the Banach space , |v| L ∞ (Ω) } and define the positive cone Lemma 4.2. For α, β ∈ (0, 1), the operator T : P → P is compact.
where G 1 (x, y) is the Green's function for (− ) α 2 in R N . In the same way, we havev According to Lemma 12.3.5 in [11], we can obtain that there exists γ 0 > 0 such thatū k is bounded in C γ0 (Ω). The embedding C γ0 (Ω) → C 0 (Ω) is compact. Then up to a subsequence of {ū k }, there existsū 0 such that u k →ū 0 , in C 0 (Ω).
Similarly, we also have that up to a subsequence of {v k }, there existsv 0 such that v k →v 0 , in C 0 (Ω).
It is easy to see that (ū 0 ,v 0 ) ∈ P . Therefore we finish the proof.