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We show the existence of locally bounded global solutions to the chemotaxissystem \[ u_t = \nabla\cdot(D(u)\nabla u) - \nabla\cdot(\frac{u}{v} \nabla v)\] \[ v_t = \Delta v - uv \] with homogeneous Neumann boundary conditions andsuitably regular positive initial data in smooth bounded domains $\Omega\subset \mathbb{R}^N$, $N\geq2$, for $D(u)\geq \delta u^{m-1}$ with some$\delta>0$, provided that $m>1+\frac N4$.

Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion