  In this paper, we study the minimizing problem: $$ S_{p,1,\alpha,\mu}:=
\inf_{u\in W^{1,p}(\mathbb{R}^{N})\setminus\{0\}} \frac{
\int_{\mathbb{R}^{N}}|\nabla u|^{p}\mathrm{d}x - \mu \int_{\mathbb{R}^{N}}
\frac{|u|^{p}}{|x|^{p}} \mathrm{d}x} {\left( \int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{p^{*}_{\alpha}}|u(y)|^{p^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x \mathrm{d}y \right)^{\frac{p}{2\cdot p^{*}_{\alpha}}}}, $$ where
$N\geqslant3$, $p\in(1,N)$, $\mu\in \left[ 0, \left( \frac{N-p}{p} \right)^{p}
\right)$, $\alpha\in(0,N)$ and $p^{*}_{\alpha}=
\frac{p}{2}\left(\frac{2N-\alpha}{N-p}\right)$ is the
Hardy--Littlewood--Sobolev upper critical exponent. Firstly, by using
refinement of Hardy-Littlewood-Sobolev inequality, we prove that
$S_{p,1,\alpha,\mu}$ is achieved in $\mathbb{R}^{N}$ by a radially symmetric,
nonincreasing and nonnegative function. Secondly, we give a estimation of
extremal function.
