In this work, we study the nonlinear homogeneous Neumann boundary value problem β(u)− diva(x,∇u)∋f in Ω,a(x,∇u)⋅η=0 on ∂Ω, where Ω is a smooth bounded open domain in ℝN,N≥3, with smooth boundary ∂Ω and η the outer unit normal vector on ∂Ω. We prove the existence and uniqueness of an entropy solution for L1-data f. The functional setting involves Lebesgue and Sobolev spaces with variable exponent

elliptic equation; variable exponent; entropy solution; L1-data; Neumann boundary condition