Stationary states of a chemotaxis consumption system with singular sensitivity and inhomogeneous boundary conditions

For given total mass $m>0$ we show unique solvability of the stationary chemotaxis-consumption model $$\{\begin{array}{l}0=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (\frac{u}{v}\mathrm{\nabla }v)\\ 0=\mathrm{\Delta }v-uv\\ {\int }_{\mathrm{\Omega }}u=m\end{array}$$ under no-flux-Dirichlet boundary conditions in bounded smooth domains $\mathrm{\Omega }\subset {\mathbb{R}}^2$ and $\mathrm{\Omega }=B_R(0)\subset {\mathbb{R}}^d$, $d\ge 3$.