  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 H\"older 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.

Analysis of PDEs