论文标题
涉及分数laplacian的系统解决方案的定性特性
Qualitative properties of solutions for system involving fractional Laplacian
论文作者
论文摘要
在本文中，我们考虑涉及分数laplacian \ begin {equion} \ left \ {\ {arnay} {array} {ll} {ll}^{s}^{s} u（x） \正确的。 （1）\ end {equation}在两种不同类型的域中，一个是有界的，另一个是无限的，其中$ 0 <s <1 $。为了研究分数方程解决方案的定性特性，常规方法是扩展方法和移动平面方法。但是，上述方法在不对称和凸形域等方面具有技术限制。在这项工作中，我们采用了分数拉普拉斯式的直接滑动方法来从不同类型的域中在$ x_n $变量中得出（1）的解决方案的单调性。同时，我们为证明中的系统开发了一种新的迭代方法，希望可以将其应用于解决其他问题。
In this paper, we consider the following nonlinear system involving the fractional Laplacian \begin{equation} \left\{\begin{array}{ll} (-Δ)^{s} u (x)= f(u,\,v), \\ (-Δ)^{s} v (x)= g(u,\,v), \end{array} \right. (1) \end{equation} in two different types of domains, one is bounded, and the other is unbounded, where $0<s<1$. To investigate the qualitative properties of solutions for fractional equations, the conventional methods are extension method and moving planes method. However, the above methods have technical limits in asymmetric and convex domains and so on. In this work, we employ the direct sliding method for fractional Laplacian to derive the monotonicity of solutions for (1) in $x_n$ variable in different types of domains. Meanwhile, we develop a new iteration method for systems in the proofs which hopefully can be applied to solve other problems.
