Strongly nonlinear nonhomogeneous elliptic unilateral problems with L^1 data and no sign conditions

In this article, we prove the existence of solutions to unilateral problems involving nonlinear operators of the form: $$ Au+H(x,u, abla u)=f $$ where $A$ is a Leray Lions operator from $W_0^{1,p(x)}(Omega)$ into its dual $W^{-1,p'(x)}(Omega)$ and $H(x,s,xi)$ is the nonlinear term satisfying some growth condition but no sign condition. The right hand side $f$ belong to $L^1(Omega)$.