Symmetry and Monotonicity of Positive Solutions to Schr\"{o}dinger Systems with Fractional $p$-Laplacian

In this paper, we first establish a narrow region principle and a decay at infinity theorem to extend the direct method of moving planes for general fractional $p$-Laplacian systems. By virtue of this method, we can investigate the qualitative properties of the following Schr"{o}dinger system with fractional $p$-Laplacian \begin{equation*} \left{\begin{array}{r@{\ \ }c@{\
}ll} \left(-\Delta\right){p}^{s}u+au^{p-1}& =&f(u,v), \[0.05cm] \left(-\Delta\right){p}^{t}v+bv^{p-1}& =&g(u,v), \end{array}\right. \end{equation*} where $0<s,t<1$ and $2<p<\mathrm{\infty }$. We obtain the radial symmetry in the unit ball or the whole space ${\mathbb{R}}^N(N\ge 2)$, the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on $f$ and $g$, respectively.