This paper addresses the existence and regularity of
weak solutions for a fully parabolic model of chemotaxis,
with prevention of overcrowding, that degenerates in a
two-sided fashion, including an extra nonlinearity
represented by a p-Laplacian diffusion term. To prove the
existence of weak solutions, a Schauder fixed-point
argument is applied to a regularized problem and the
compactness method is used to pass to the limit. The
local Hölder regularity of weak solutions is established
using the method of intrinsic scaling. The results are a
contribution to showing, qualitatively, to what extent
the properties of the classical Keller--Segel chemotaxis
models are preserved in a more general setting. Some
numerical examples illustrate the model.