Qualitative properties of solutions for system involving fractional Laplacian

TL;DR

This paper studies the qualitative properties of solutions to a nonlinear fractional Laplacian system in various domains, introducing a direct sliding method and a new iteration technique to overcome limitations of traditional methods.

Contribution

It develops a direct sliding method for fractional Laplacian systems and introduces a novel iteration approach, expanding analytical tools for asymmetric and convex domains.

Findings

Established monotonicity of solutions in different domains.

Overcame limitations of extension and moving planes methods.

Proposed a new iteration method applicable to other fractional problems.

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.