Variable exponent Hardy-type inequalities in $\mathbb{R}^n$

Abstract

In this paper, we investigate further the weighted $p(x)$-Hardy inequality with the additional term of the form \[ \int_Ω|ξ|^{p(x)}μ_{1,β} (dx) \leqslant \int_Ω|\nabla ξ|^{p(x)}μ_{2,β} (dx)+\int_Ω\left|ξ{\log ξ} \right|^{p(x)} μ_{3,β} (dx), \] holding for Lipschitz functions compactly supported in $Ω\subseteq\mathbb{R}^n$. The involved measures depend on a certain solution to the partial differential inequality involving $p(x)$-Laplacian ${-}Δ_{p(x)} u\geqslant Φ$, where $Φ$ is a given locally integrable function, and $u$ is defined on an open and not necessarily bounded subset $Ω\subseteq\mathbb{R}^n $, and a certain parameter $β$. We focus on the $n$-dimensional case giving some examples. Moreover, we compare our inequalities with the existing in the literature.