In this paper we will prove the existence of solutions of the unilateral problem
Au - div Φ(x,u) + H(x, u, ∇ u) = μ
in Musielak spaces, where A is a Leray-Lions operator defined on D(A) ⊂ W01 LM (Ω), μ ∈ L(Ω) + W-1 EM'(Ω), where M and M' are two complementary Musielak-Orlicz functions and both the first and the second lower terms Φ and H satisfies only the growth condition and u ≥ ζ where ζ is a measurable function.
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