Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions

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.