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

الأصل · EN

We derive a family of weighted Hardy-type inequalities in the variable exponent Lebesgue space with an additional term of the form ∫Ω |ξ|ᵖ⁽ˣ⁾ μ₁,ᵦ(dx) ∫Ω|∇ ξ|ᵖ⁽ˣ⁾μ₂,ᵦ(dx)+∫Ω|ξ ξ |ᵖ⁽ˣ⁾ μ₃,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 -Δₚ₍ₓ₎ u Φ, where Φ is a given locally integrable function, and u is defined on an open and not necessarily bounded subset Ωⁿ, 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.