Given \(0<s<1\), \(0<\beta<n\), \(p>2\) and \(q>\frac{p}{2}\) this article studies geometric properties of positive solutions to the following type of Choquard equation \[ (-\Delta)^s_pu=\Big( \frac{1}{|x|^{n-\beta}}\ast u^q\Big)u^{q-1}\text{ in }\mathbb{R}^n,\] where \((-\Delta)^s_p\) denotes the fractional \(p\)-Laplacian. To be precise, assuming that a solution \(u\in C^{1,1}_{\mathrm{loc}}(\mathbb{R}^n)\) of the above equation satisfies additionally the integrability assumption \[ \int_{\mathbb{R}^n}\frac{|1+u(x)|^{p-1}}{1+|x|^{n+sp}}\ dx<\infty \] and, for some \(\gamma\) such that \(\min\{\gamma(q-p)+n,\gamma(2q-p)\}>sp+\beta\), \[ u(x) \sim\frac{1}{|x|^{\gamma}}\text{ as }|x|\to\infty,\] the author shows that \(u\) must be radially symmetric and monotone decreasing about some point in \(\mathbb{R}^n\). In this way, the result is a generalization to [L. Ma and Z. Zhang, Nonlinear Anal. 182, 248–262 (2019; Zbl 1411.35273)], where it was necessary to assume \(0<\beta<2\) and \(q>p-1\). As noted by the author, it follows that \(\gamma\) must be positive and it is enough to assume \(u\) is nonnegative.
To prove the main result, the author applies the moving plane method as described by W. Chen and C. Li [Adv. Math., 335, 735–758 (2018; Zbl 1395.35055)] and uses in particular certain maximum principles and boundary estimates for the fractional \(p\)-Laplacian from this reference.