Strongly Nonlinear Elliptic Unilateral Problems without Sign Condition and L ¹ Data

We prove the existence of solutions of unilateral problems involving nonlinear operators of the form

where

A A

is a Leray Lions operator from

W_0^{1, p}(\Omega) W 0 1 , p ( Ω )

into its dual

W^{-1, p'}(\Omega) W − 1 , p ′ ( Ω )

and

H(x, u, \nabla u) H ( x , u , ∇ u )

is a nonlinearity which satisfies the following growth condition

|H(x, s, \xi)| \leq \gamma(x)+g(s) |\xi|^p ∣ H ( x , s , ξ ) ∣ ≤ γ ( x ) + g ( s ) ∣ ξ ∣ p

with

\gamma\in L^1(\Omega) γ ∈ L 1 ( Ω )

and

g\in L^1({\mathbb R}) g ∈ L 1 ( R )

, and without assuming any sign condition on

H(x, s, \xi) H ( x , s , ξ )

. The right hand side

f f

belongs to

L^1(\Omega) L 1 ( Ω )

.