  We investigate the parabolic-elliptic Keller-Segel model
\begin{align*}\left\{\begin{array}{r@{\,}l@{\quad}l@{\quad}l@{\,}c}
u_{t}&=\Delta u-\,\chi\nabla\!\cdot(\frac{u}{v}\nabla v),\ &x\in\Omega,& t>0,\\
0&=\Delta v-\,v+u,\ &x\in\Omega,& t>0,\\ \frac{\partial
u}{\partial\nu}&=\frac{\partial v}{\partial\nu}=0,\ &x\in\partial\Omega,&
t>0,\\ u(&x,0)=u_0(x),\ &x\in\Omega,& \end{array}\right. \end{align*} in a
bounded domain $\Omega\subset\mathbb{R}^n$ $(n\geq2)$ with smooth boundary.
  \noindent We introduce a notion of generalized solvability which is
consistent with the classical solution concept, and we show that whenever
$0<\chi<\frac{n}{n-2}$ and the initial data satisfy only certain requirements
on regularity and on positivity, one can find at least one global generalized
solution.
