Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

Abstract

We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving A-Laplacian −Δ A u = −divA(∇u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset $${\Omega \subseteq \mathbb{R}^n}$$ . Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the form $$\int_\Omega F_{\bar{A}}(|\xi|) \mu_1(dx) \leq \int_\Omega \bar{A}(|\nabla \xi|)\mu_2(dx),$$ where $${\bar{A}(t)}$$ is a Young function related to A and satisfying Δ′-condition, while $${F_{\bar{A}}(t) = 1/(\bar{A}(1/t))}$$ . Examples involving $${\bar{A}(t) = t^p{\rm log}^\alpha(2+t), p \geq 1, \alpha \geq 0}$$ are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30–50, 2013), where we dealt with inequality −Δ p u ≥ Φ, leading to Hardy and Hardy–Poincare inequalities with the best constants.