Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form

-∑i=1N∂iai(x,u,∇u)-∑i=1N∂iφi(u)=f,$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$

where the right hand side f belongs to L¹(Ω). The operator -∑i=1N∂iai(x,u,∇u)$- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $ is a Leray-Lions anisotropic operator and ϕ_i ∈ C⁰(ℝ,ℝ).