We study in this paper the anisotropic nonlinear boundary value problem
N ∂  ∂u
− ∑∂x ai x,∂x =μ in Ω, u=0 on ∂Ω, where Ω is a smooth bounded open
i=1i i
domain in RN , N ≥ 3 and μ a bounded Radon measure. We prove the existence of a weak energy solution for this anisotropic nonlinear elliptic problem with different vari- able exponents, so that, the functional setting involves anisotropic variable exponent Sobolev spaces and Marcinkiewicz spaces.

Generalized Lebesgue-Sobolev spaces, anisotropic Sobolev spaces, weak solution, bounded Radon measure, electrorheological fluids, Marcinkiewicz spaces