We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consideron a Lipschitz bounded domain in R N . The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N -function M . The approach does not require any particular type of growth condition of M or its conjugate M * (neither ∆ 2 , nor ∇ 2 ). The condition we impose is log-Hölder continuity of M , which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.

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Abstract. We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving ALaplacian −ΔAu = −divA(∇u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset Ω ⊆ 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 formwhereĀ(t) is a Young function related to A and satisfying Δ -condition, while FĀ(t) = 1/(Ā(1/t)). Examples involvingĀ(t) = t p log α (2+t), p ≥ 1, α ≥ 0 are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30-50, 2013), where we dealt with inequality −Δpu ≥ Φ, leading to Hardy and Hardy-Poincaré inequalities with the best constants. Mathematics Subject Classification (2010). Primary 26D10; Secondary 31B05, 35D30, 35J60, 35R45.