The minimizing problem involving p--Laplacian and Hardy--Littlewood--Sobolev upper critical exponent

Abstract

In this paper, we study the minimizing problem: $$S_{p,1,\alpha ,\mu }:=\underset{u\in W^{1,p}({\mathbb{R}}^N)\setminus \{0\}}{\inf }\frac{{\int }_{{\mathbb{R}}^N}|\nabla u{|}^p\mathrm{d}x-\mu {\int }_{{\mathbb{R}}^N}\frac{|u{|}^p}{|x{|}^p}\mathrm{d}x}{{\left({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x){|}^{p_{\alpha }^{\ast }}|u(y){|}^{p_{\alpha }^{\ast }}}{|x-y{|}^{\alpha }}\mathrm{d}x\mathrm{d}y\right)}^{\frac{p}{2\cdot p_{\alpha }^{\ast }}}},$$ where $N⩾3$, $p\in (1,N)$, $\mu \in \left[0,{\left(\frac{N-p}{p}\right)}^p\right)$, $\alpha \in (0,N)$ and $p_{\alpha }^{\ast }=\frac{p}{2}\left(\frac{2N-\alpha }{N-p}\right)$ 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.