  We consider the question of existence of weak solutions for the fully
inhomogeneous, stationary generalized Navier-Stokes equations for homogeneous,
shear-thinning fluids. For a shear rate exponent $p \in \big(\tfrac{2d}{d+1},
2\big)$, previous results require either smallness of the norm or vanishing of
the normal component of the boundary data. In this work, combining previous
methods, we propose a new, more general smallness condition.
