Existence of solutions for nonlinear degenerate elliptic equations with Lm-data and Neumann boundary condition

In this paper we consider the following homogenous Neumann elliptic problem {cases} -∑ᵢ₌₁N D^i(a_i(x, u, ∇ u)) + ∑ᵢ₌₁N A(|u|)D^i up_i + |u|p_0-2u = f & in Ω, \\ ∑ᵢ₌₁N a_i(x, u, ∇ u) · n_i = 0 & on ∂Ω, {cases} where Ω is a bounded open domain in R^N ( N ≥ 2 ) with Lipschitz boundary ∂Ω , we prove the existence of solutions in the sense of distributions for our elliptic problem in the anisotropic Sobolev spaces.