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 { 0 = Δ u − χ ∇ ⋅ ( u v ∇ v ) 0 = Δ v − u v ∫ Ω u = m under no-flux-Dirichlet boundary conditions in bounded smooth domains Ω ⊂ R 2 and Ω = B R ( 0 ) ⊂ R d , d ≥ 3 .