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 \[ \int_\Omega |\xi|^{p(x)}\mu_{1,\beta} (dx) \leqslant \int_\Omega |\nabla \xi|^{p(x)}\mu_{2,\beta} (dx)+\int_\Omega \left|\xi{\log \xi} \right|^{p(x)} \mu_{3,\beta} (dx), \] holding for Lipschitz functions compactly supported in $\Omega\subseteq\mathbb{R}^n$. The involved measures depend on a certain solution to the partial differential inequality involving $p(x)$-Laplacian ${-}\Delta_{p(x)} u\geqslant \Phi$, where $\Phi$ is a given locally integrable function, and $u$ is defined on an open and not necessarily bounded subset $\Omega\subseteq\mathbb{R}^n $, and a certain parameter $\beta$. We focus on the $n$-dimensional case giving some examples. Moreover, we compare our inequalities with the existing in the literature.