The paper deals with the inhomogeneous nonlinear Neumann boundary value problem
−div(Φ(∇u−Θ(u)))+|u|p(x)−2u+α(u)=finΩ,

(Φ(∇u−Θ(u))⋅η+γ(u)=gonΩ
with
Φ(ξ)=|ξ|p(x)−2ξ,∀ξ∈ℝN,
where Ω⊆ℝN (N≥3) is a bounded open domain with Lipschitz boundary ∂Ω, η is the outer unit normal vector on ∂Ω, α, γ, Θ are real functions defined on ℝ of ℝN, f∈L1(Ω), g∈L1(∂Ω) and p:Ω¯→ℝ is a continuous function such that 1

generalized Sobolev spaces; Neumann boundary conditions; entropy solution; noncoercive operator