Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity

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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ $(n\ge 2)$ 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.