In this work, we study the following nonlinear non-homogeneous Fourier boundary value problem b(u)−div(a(x,∇u))=f in Ω, a(x,∇u)⋅η+λu=g on ∂Ω, where Ω is a smooth bounded open domain in ℝN, N≥3, p∈C(Ω). We prove the existence and uniqueness of a weak solution for f∈L∞(Ω) and g∈L∞(∂Ω), the existence and uniqueness of an entropy solution for L1-data f and g. The functional setting involves Lebesgue and Sobolev spaces with variable exponents

Lebesgue spaces with variable exponent; Sobolev spaces with variable exponent; weak solution; entropy solution; Fourier-type boundary condition