In this paper we consider the following fractional system $$\begin{array}{} \displaystyle \left\{ \begin{gathered} F(x,u(x),v(x),{\mathcal{F}_\alpha }(u(x))) = 0,\\ G(x,v(x),u(x),{\mathcal{G}_\beta }(v(x))) = 0, \\ \end{gathered} \right. \end{array}$$ where 0 < α, β < 2, 𝓕α and 𝓖β are the fully nonlinear fractional operators: $$\begin{array}{} \displaystyle {\mathcal{F}_\alpha }(u(x)) = {C_{n,\alpha }}PV\int_{{\mathbb{R}^n}} {\frac{{f(u(x) - u(y))}} {{{{\left| {x - y} \right|}^{n + \alpha }}}}dy} ,\\ \displaystyle{\mathcal{G}_\beta }(v(x)) = {C_{n,\beta }}PV\int_{{\mathbb{R}^n}} {\frac{{g(v(x) - v(y))}} {{{{\left| {x - y} \right|}^{n + \beta }}}}dy} . \end{array}$$ A decay at infinity principle and a narrow region principle for solutions to the system are established. Based on these principles, we prove the radial symmetry and monotonicity of positive solutions to the system in the whole space and a unit ball respectively, and the nonexistence in a half space by generalizing the direct method of moving planes to the nonlinear system.

In this article, we consider the cooperative semi-linear fractional system (−∆) α 2 u(x) = h(x, u(x)), where 0 < α < 2, u and h stand for k-dimentional vector-valued functions, and h(x, u(x)) is locally Lipschitz in u. We first establish two narrow region principles for different cases. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions of the above system in bounded star-shaped domains and the whole space.

The aim of this article is to consider the semi-linear fractional system with Sobolev exponents q = n+α n-β and p = n+β n-α (α = β): (-) α/2 u(x) = k(x)v q (x) + f (v(x)), (-) β/2 v(x) = j(x)u p (x) + g(u(x)), where 0 < α, β < 2. We first establish two maximum principles for narrow regions in the ball and out of the ball by the iteration technique, respectively. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions to the above system in the whole space and bounded star-shaped domain. As a consequence, the monotonic decreasing properties of W(x) = |x| n-α 2 u(x) and W 1 (x) = |x| n-β 2 v(x) along the radial direction in the whole space are obtained.