Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to weighted doubly $D^{1,p}$-critical quasi-linear nonlocal elliptic equations with Hardy potential

In this paper, we mainly consider nonnegative weak solutions $u\in D^{1,p}(\R^{N})$ to the doubly $D^{1,p}(\R^{N})$-critical nonlocal quasi-linear Schr\"{o}dinger-Hartree equation: \begin{align*} -\Delta_p u- \mu \frac{u^{p-1}}{|x|^p}=\left(|x|^{-2p}\ast |u|^{p}\right)|u|^{p-2}u \qquad &\mbox{in} \,\, \mathbb{R}^N, \end{align*} where $N\geq3$, $0\leq\mu< \bar{\mu}:=\left( (N-p)/p \right)^p$ and $1<p<\frac{N}{2}$. When $\mu>0$, due to appearance of the Hardy potential, the equation has singularity at $0\in\mathbb{R}^{N}$ and hence is not translation invariant, so sharp asymptotic estimates near the origin must be involved. First, we establish regularity and the sharp estimates on asymptotic behaviors near the origin and the infinity for any positive solution $u\in D^{1,p}(\R^{N})$ (and $|\nabla u|$) to more general equation $-\triangle_p u - \mu \frac{1}{|x|^p}u^{p-1}=V(x)\frac{1}{|x|^s}u^{p-1}$ with $N\geq2$, $0\leq\mu< \bar{\mu}$, $1<p<N$, $0\leq s < p$ and $0\leq V(x)\in L^\frac{N}{p-s}(\R^N)$. Then, as a consequence, we can apply the method of moving planes to prove that all the nontrivial nonnegative solutions in $D^{1,p}(\R^{N})$ are radially symmetric and strictly radially decreasing about the origin $0\in\mathbb{R}^{N}$. The sharp asymptotic estimates and radial symmetry for more general weighted doubly $D^{1,p}$-critical nonlocal quasi-linear equations were also derived. Our results extend the results in \cite{DLL} from the special case $\mu=0$ to general cases $0\leq\mu<\bar{\mu}$.