Abstract In this paper, we will study the existence of solutions for some nonlinear anisotropic elliptic equation of the type { A u + g ( x , u , ∇ u ) = μ − d i v φ ( u ) i n Ω , u = 0 o n ∂ Ω , \left\{ {\matrix{{Au + g\left( {x,u,\nabla u} \right) = \mu - div\,\phi \left( u \right)} \hfill & {in\,\Omega ,} \hfill \cr {u = 0} \hfill & {on\,\,\partial \Omega ,} \hfill \cr } } \right. where A u = − ∑ i = 1 N ∂ ∂ x i a i ( x , u , ∇ u ) Au = - \sum\limits_{i = 1}^N {{\partial \over {\partial {x_i}}}{a_i}\left( {x,u,\nabla u} \right)} is a Leray-Lions operator, the Carathéodory function g(x, s, ξ) is a nonlinear lower order term that verify some natural growth and sign conditions, where the data µ = f − div F belongs to L 1−dual and ϕ (·) ∈ C 0(R, R N ).