In this paper, we prove the existence of solutions for some strongly nonlinear Dirichlet problems whose model is the following −div(M−1 M(|∇u|) ∇u |∇u| ) + uM(|∇u|) = f − divF in D0 (Ω), where Ω is an open bounded subset of IRN , N ≥ 2. We emphasize that no ∆2-condition is required for the N-function M.