Multipolar Hardy inequalities and mutual interaction of the poles

Abstract

In this paper we state the weighted Hardy inequality \begin{equation*} c\int_{{\mathbb R}^N}\sum_{i=1}^n \frac{φ^2 }{|x-a_i|^2}\, μ(x)dx\le \int_{{\mathbb R}^N} |\nablaφ|^2 \, μ(x)dx +k \int_{\mathbb{R}^N}φ^2 \, μ(x)dx \end{equation*} for any $\phi$ in a weighted Sobolev spaces, with $c\in ]0,c_o[$ where $c_o=c_o(N,\mu )$ is the optimal constant, $a_1,\dots ,a_n\in {\mathbb{R}}^N$, $k$ is a constant depending on $\mu$. We show the relation between $c$ and the closeness to the single pole. To this aim we analyze in detail the difficulties to be overcome to get the inequality.