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Qualitative properties of solutions for system involving the fractional Laplacian
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 26 February 2024, pp. 1-16
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In this paper, we consider the following non-linear system involving the fractional Laplacian0.1
\begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation}in two different types of domains, one is bounded, and the other is an infinite cylinder, where $0< s<1$
. We employ the direct sliding method for fractional Laplacian, different from the conventional extension and moving planes methods, to derive the monotonicity of solutions for (0.1) in $x_n$
variable. Meanwhile, we develop a new iteration method for systems in the proofs. Hopefully, the iteration method can also be applied to solve other problems.
