  Previous studies of chemotaxis models with consumption of the chemoattractant
(with or without fluid) have not been successful in explaining pattern
formation even in the simplest form of concentration near the boundary, which
had been experimentally observed.
  Following the suggestions that the main reason for that is usage of
inappropriate boundary conditions, in this article we study solutions to the
stationary chemotaxis system \[
  \begin{cases}
  0 = \Delta n - \nabla\cdot(n\nabla c) \\
  0 = \Delta c - nc
  \end{cases} \] in bounded domains $\Omega\subset\mathbb{R}^N$, $N\ge 1$,
under no-flux boundary conditions for $n$ and the physically meaningful
condition \[
  \partial_{\nu} c = (\gamma-c)g \] on $c$, with given parameter $\gamma>0$ and
$g\in C^{1+\beta}(\Omega)$ satisfying $g\ge 0$, $g \not\equiv 0$ on $\partial
\Omega$. We prove existence and uniqueness of solutions for any given mass
$\int_\Omega n > 0$. These solutions are non-constant.
