Global existence and boundedness of solutions to a chemotaxis-consumption model with singular sensitivity

In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_t=\Delta u -\chi \nabla \cdot (\frac{u}{v} \nabla v) \\ v_t=\Delta v-f(u)v \end{cases} \end{equation*} in a smooth and bounded domain $\Omega$ of $\mathbb{R}^2$, with $\chi>0$ and $f\in C^1(\mathbb{R})$ essentially behaving like $u^\beta$, $0<\beta<1$. Precisely for $\chi<1$ and any sufficiently regular initial data $u(x,0)\geq 0$ and $v(x,0)>0$ on $\bar{\Omega}$, we show the existence of global classical solutions. Moreover, if additionally $m:=\int_\Omega u(x,0)$ is sufficiently small, then also their boundedness is achieved.