| Summary: | 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 $N\geqslant3$, $p\in(1,N)$, $\mu\in\big[0,\big(\frac{N-p}{p}\big)^{p} \big)$, $\alpha\in(0,N)$ and $p^{*}_{\alpha}=\frac{p}{2}\big(\frac{2N-\alpha}{N-p}\big)$ 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.
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