Multipolar Hardy inequalities in $L^{p}$-spaces

Abstract

The aim of this paper is twofold. On the one hand, we establish with two different methods the new multipolar Hardy inequality

$$\begin{aligned}&\frac{(N-2)^{2}}{np^{2}}\mathop \int \limits _{{\mathbb {R}}^{N}}\sum _{i=1}^{n}\frac{|u|^{p}}{|x-a_{i}|^{2}}dx\\&\qquad +\frac{(N-2)^{2}}{2p^{2}n^{2}}\mathop \int \limits _{{\mathbb {R}}^{N}}\sum _{\begin{array}{c} i,j=1\\ i\ne j \end{array}}^{n}\frac{|a_{i}- a_{j}|^{2}}{|x-a_{i}|^{2}|x-a_{j}|^{2}}|u|^{p}dx\\&\quad \le \mathop \int \limits _{{\mathbb {R}}^{N}}|\nabla u|^{2}|u|^{p-2}dx \end{aligned}$$

for every $u \in H^{1,p}({\mathbb {R}}^{N})$, $p\ge 2$, $a_{1}, a_{2},\ldots , a_{n} \in {\mathbb {R}}^{N}$, $N\ge 3$, and $n\ge 2$. On the other hand, we prove the weighted multipolar Hardy inequality

$$\begin{aligned}&\frac{(N+k_{2}-2)^{2}}{np^{2}}\mathop \int \limits _{{\mathbb {R}}^{N}} \sum _{i=1}^{n}\frac{|u|^{p}}{|x-a_{i}|^{2}}d\mu \\&\qquad +\frac{(N+k_{2}-2)^{2}}{2n^{2}p^{2}}\mathop \int \limits _{{\mathbb {R}}^{N}}\sum _{\begin{array}{c} i,j=1\\ i\ne j \end{array}}^{n}\frac{|a_{i}- a_{j}|^{2}}{|x-a_{i}|^{2}|x-a_{j}|^{2}}|u|^{p}d\mu \nonumber \\&\quad \le \mathop \int \limits _{{\mathbb {R}}^{N}}|\nabla u|^{2}|u|^{p-2}d\mu +k_{1}\mathop \int \limits _{{\mathbb {R}}^{N}}|u|^{p}d\mu \end{aligned}$$

for every $u \in H^{1,p}({\mathbb {R}}^{N},d\mu )$, $d\mu =\mu (x)dx$, $p\ge 2$, $a_{1}, a_{2},\ldots , a_{n} \in {\mathbb {R}}^{N}$, $N\ge 3$, $n\ge 1$, and some constants $k_{1}, k_{2} \in {\mathbb {R}}$, $k_{2}>2-N$. The weighted functions $\mu$ are of a quite general type.