Long-term behaviour in a parabolic-elliptic chemotaxis-consumption model

Global existence and boundedness of classical solutions of the chemotaxis–consumption system \begin{align*} n_t &= \Delta n - \nabla \cdot (n \nabla c), \ 0 &= \Delta c - nc, \end{align*} under no-flux boundary conditions for $n$ and Robin-type boundary conditions $${\mathrm{\partial }}_{\nu }c=(\gamma -c)g$$ for $c$ (with $\gamma >0$ and $C^{1+\beta }(\mathrm{\partial }\mathrm{\Omega })\ni g>0$ for some $\beta \in (0,1)$) are established in bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 1$. Under a smallness condition on $\gamma$, moreover, we show convergence to the stationary solution.