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 $${\int }_{\mathrm{\Omega }}\mathrm{\mid }\xi {\mathrm{\mid }}^{p(x)}{\mu }_{1,\beta }(dx)⩽{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }\xi {\mathrm{\mid }}^{p(x)}{\mu }_{2,\beta }(dx)+{\int }_{\mathrm{\Omega }}{\mid \xi \log \xi \mid }^{p(x)}{\mu }_{3,\beta }(dx),$$ 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$\Phi$isagivenlocallyintegrablefunction,and$u$ is defined on an open and not necessarily bounded subset $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, 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.