  In this paper, we study the existence of weak solutions to a steady system
that describes the motion of a micropolar electrorheological fluid. The
constitutive relations for the stress tensors belong to the class of
generalized Newtonian fluids. The analysis of this particular problem leads
naturally to weighted variable exponent Sobolev spaces. We establish the
existence of solutions for a material function $\hat p$ that is
$\log$--H\"older continuous and an electric field $\mathbf{E}$ for that
$\vert\mathbf{E}\vert^2$ is bounded and smooth. Note that these conditions do
not imply that the variable shear exponent $p=\hat p\circ\vert
\mathbf{E}\vert^2$ is globally $\log$--H\"older continuous.
