  In this paper, we mainly consider nonnegative weak solution to the
$D^{1,p}(\R^{N})$-critical quasi-linear static Schr\"{o}dinger-Hartree equation
with $p$-Laplacian $-\Delta_{p}$ and nonlocal nonlinearity: \begin{align*}
-\Delta_p u =\left(|x|^{-2p}\ast |u|^{p}\right)|u|^{p-2}u \qquad &\mbox{in}
\,\, \mathbb{R}^N, \end{align*} where $1<p<\frac{N}{2}$, $N\geq3$ and $u\in
D^{1,p}(\R^N)$. Being different to the $D^{1,p}(\R^{N})$-critical local
nonlinear term $u^{p^{\star}-1}$ with $p^{\star}:=\frac{Np}{N-p}$ investigated
in \cite{CFR,LDSMLMSB,GV,Ou,BS16,VJ16} etc., since the nonlocal convolution
$|x|^{-2p}*u^p$ appears in the Hartree type nonlinearity, it is impossible for
us to use the scaling arguments and the Doubling Lemma as in \cite{VJ16} to get
preliminary estimates on upper bounds of asymptotic behaviors for any positive
solutions $u$. Moreover, it is also quite difficult to obtain the boundedness
of the quasi-norm $\|u \|_{L^{s,\infty}(\R^N)}$ and hence derive the sharp
estimates on upper bounds of asymptotic behaviors from the preliminary
estimates as in \cite{VJ16}. Fortunately, by showing a better preliminary
estimates on upper bounds of asymptotic behaviors through the De
Giorgi-Moser-Nash iteration method and combining the result from \cite{XCL}, we
are able to overcome these difficulties and establish regularity and the sharp
estimates on both upper and lower bounds of asymptotic behaviors for any
positive solution $u$ to more general equation $-\Delta_p u=V(x)u^{p-1}$ with
$V\in L^{\frac{N}{p}}(\mathbb{R}^{N})$. Then, by using the arguments from
\cite{BS16,VJ16}, we can deduce the sharp estimates on both upper and lower
bounds for the decay rate of $|\nabla u|$. Finally, as a consequence, we can
apply the method of moving planes to prove that all the nontrivial nonnegative
solutions are radially symmetric and strictly decreasing about some point
$x_0\in\R^N$.
