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 \[ {cases} 0= Δ u - χ ∇ · (u/v ∇ v) \\ 0= Δ v - uv \\ ∫_Ω u = m {cases} \] under no-flux-Dirichlet boundary conditions in bounded smooth domains Ω⊂ R² and Ω=BR(0)⊂ Rᵈ, d≥ 3.