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 $\mathrm{\Omega }$ 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 $u(x,0)\ge 0$ and $v(x,0)>0$ on $\bar{\mathrm{\Omega }}$, we show the existence of global classical solutions. Moreover, if additionally $m:={\int }_{\mathrm{\Omega }}u(x,0)$ is sufficiently small, then also their boundedness is achieved.