Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

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–Poincaré inequalities with the best constants.