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 $\Omega \subseteq {\mathbb{R}}^n$. The involved measures depend on a certain solution to the partial differential inequality involving $p(x)$-Laplacian $-{\Delta }_{p(x)}u⩾\Phi$, 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.

18 pages. arXiv admin note: substantial text overlap with arXiv:1407.6226

References in corpus (1)

• Hardy type inequality in variable Lebesgue spaces