Strongly Nonlinear Elliptic Unilateral Problems without Sign Condition and L¹ Data

We prove the existence of solutions of unilateral problems involving nonlinear operators of the form $$Au + H(x, u, \nabla u) = f$$ where $A$ is a Leray Lions operator from $W_0^{1, p}(\Omega)$ into its dual $W^{-1, p'}(\Omega)$ and $H(x, u, \nabla u)$ is a nonlinearity which satisfies the following growth condition $|H(x, s, \xi)| \leq \gamma(x)+g(s) |\xi|^p$ with $\gamma\in L^1(\Omega)$ and $g\in L^1({\mathbb R})$, and without assuming any sign condition on $H(x, s, \xi)$. The right hand side $f$ belongs to $L^1(\Omega)$.