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Positive solutions for a weighted fractional system. (English) Zbl 1438.35256

Summary: In this article, we study positive solutions to the system \[\begin{cases} \mathcal{A}_\alpha u(x) = C_{n,\alpha} PV \int_{R^n} \frac{a_1(x-y)(u(x)-u(y))}{|x-y|^{n+\alpha}}\mathrm{d}y = f(u(x),v(x)), \\ \mathcal{B}_\beta v(x) = C_{n,\beta} PV \int_{R^n} \frac{a_2(x-y)(v(x)-v(y))}{|x-y|^{n+\beta}} \mathrm{d}y= g(u(x),v(x)). \end{cases}\] To reach our aim, by using the method of moving planes, we prove a narrow region principle and a {decay at infinity} by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.

MSC:

35K99 Parabolic equations and parabolic systems
35B09 Positive solutions to PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
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