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 $$\int_\Omega\ |\xi|^{p(x)} \mu_{1,\beta}(dx)\leqslant \int_\Omega |\nabla \xi|^{p(x)}\mu_{2,\beta}(dx)+\int_\Omega \left|\xi{\log \xi} \right|^{p(x)} \mu_{3,\beta}(dx),$$ where $\xi$ is any compactly supported Lipschitz function. 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 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.