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

Abstract

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