  The chemotaxis system
  \begin{align*} u_t &= \Delta u - \nabla \cdot (u\nabla v), \\ v_t &= \Delta v
- uv,
  \end{align*} is considered under the boundary conditions $\frac{\partial
u}{\partial\nu}- u\frac{\partial v}{\partial\nu}=0$ and $v=v_\star$ on
$\partial\Omega$, where $\Omega\subset\mathbb{R}^n$ is a ball and $v_\star$ is
a given positive constant.
  In the setting of radially symmetric and suitably regular initial data, a
result on global existence of bounded classical solutions is derived in the
case $n=2$, while global weak solutions are constructed when $n\in \{3,4,5\}$.
This is achieved by analyzing an energy-type inequality reminiscent of global
structures previously observed in related homogeneous Neumann problems.
Ill-signed boundary integrals newly appearing therein are controlled by means
of spatially localized smoothing arguments revealing higher order regularity
features outside the spatial origin.
  Additionally, unique classical solvability in the corresponding stationary
problem is asserted, even in nonradial frameworks.
