We study the following nonlinear elliptic boundary value problem
b(u)−diva(x,∇u)=fa(x,∇u).η=−|u|p(x)−2u in Ω, on ∂Ω,
where Ω is a smooth bounded open domain in ℝN,N≥1 with smooth boundary ∂Ω. We prove the existence and uniqueness of a weak solution for f∈L∞(Ω), the existence and uniqueness of an entropy solution for L1-data f. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.

Lebesgue and Sobolev spaces with variable exponent; weak solution; entropy solution; Robin type boundary condition