  This paper is devoted to the analysis of non-negative solutions for a
degenerate parabolic-elliptic Patlak-Keller-Segel system with critical
nonlinear diffusion in a bounded domain with homogeneous Neumann boundary
conditions. Our aim is to prove the existence of a global weak solution under a
smallness condition on the mass of the initial data, there by completing
previous results on nite blow-up for large masses. Under some higher regularity
condition on solutions, the uniqueness of solutions is proved by using a
classical duality technique.
