Solvability of Some Strongly Unilateral Problems in L1 without Regularity Condition on the Obstacle

Consider the following nonlinear Dirichlet problem:( Au = −div(a(x, u,∇u)))+ g(x, u,∇u) = f ∈ L1(Ω), where g is a nonlinearity satisfying the following growth: |g(x, s, ξ) | ≤ b(|s|)[h(x)+|ξ|p] and the sign condition. Our objective is to study the existence result for the following unilateral problems, Tk(u) ∈W 1,p0 (Ω), u ≥ ψ a.e. in Ω, g(x, u,∇u) ∈ L1(Ω),∫ Ω a(x, u,∇u)∇Tk(u − v) dx