Existence results for some unilateral problems without sign condition with obstacle free in Orlicz spaces

We prove the existence results in the setting of Orlicz spaces for the unilateral problem associated to the following equation, A u + g ( x , u , ∇ u ) = f , where A is a Leray–Lions operator acting from its domain D ( A ) ⊂ W 0 1 L M ( Ω ) into its dual, while g ( x , u , ∇ u ) is a nonlinear term having a growth conditions with respect to ∇ u and no growth with respect to u , but does not satisfy any sign condition. The right-hand side f belongs to L 1 ( Ω ) , and the obstacle is a measurable function.