This paper is devoted to study some nonlinear elliptic Neumann equations of the type
⎧⎩⎨⎪⎪Au+g(x,u,∇u)+|u|q(⋅)−2u=f(x,u,∇u)∑i=1Nai(x,u,∇u)⋅ni=0in Ω,on ∂Ω,
in the anisotropic variable exponent Sobolev spaces, where A is a Leray-Lions operator and g(x,u,∇u), f(x,u,∇u) are two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem

quasilinear elliptic equation; Neumann problem; existence