  In this work we investigate the existence of weak solutions for steady flows
of generalized incompressible and homogeneous viscous fluids. The problem is
modeled by the steady case of the generalized Navier-Stokes equations, where
the exponent $q$ that characterizes the flow depends on the space variable:
$q=q(\mathbf{x})$. For the associated boundary-value problem we prove the
existence of weak solutions for any variable exponent
$q\geq\alpha>\frac{2N}{N+2}$, where $\alpha=\mathrm{ess}\inf q$. This work
improves all the known existence results in the sense that the lowest possible
bound of $q$ is attained and no other assumption on the regularity of $q$ is
required.
