Qualitative properties of solutions for system involving fractional Laplacian

Abstract

In this paper, we consider the following nonlinear system involving the fractional Laplacian \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{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.