Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider−divA(x,∇u)=f∈L1(Ω), on a Lipschitz bounded domain in RN. 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 Δ2, 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.