Hardy-type inequality in variable exponent Lebesgue spaces derived from nonlinear problem

We derive a family of weighted Hardy-type inequalities in the variable exponent Lebesgue space with an additional term of the form $\int_Ω |ξ|^{p(x)} μ_{1,β}(dx)\leqslant \int_Ω|\nabla ξ|^{p(x)}μ_{2,β}(dx)+\int_Ω\left|ξ{\log ξ} \right|^{p(x)} μ_{3,β}(dx),$ where $ξ$ is any compactly supported Lipschitz function. 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 derive new Caccioppoli-type inequality for the solution $u$. As its consequence we get Hardy-type inequality. We illustrate the result by several one-dimensional examples.

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