Symmetry of Positive Solutions to Choquard Type Equations Involving the Fractional $p$-Laplacian

We study symmetric properties of positive solutions to the Choquard type equationwhere $0<s<1$, $0<\alpha <n$, $p\ge 2$, $q>1$, $r>0$, $a\ge 0$ and ${(-\mathrm{\Delta })}_p^s$ is the fractional $p$-Laplacian. Via a direct method of moving planes, we prove that every positive solution $u$ which has an appropriate decay property must be radially symmetric and monotone decreasing about some point, which is the origin if $a>0$.