  This paper is concerned with the following singularly perturbed non-local
semi-linear problem \begin{equation} \label{h} \tag{$\ast$} \begin{cases}
\varepsilon^2 \Delta u=\frac{m}{\int_{\Omega}e^{u}{\mathrm{d}x}}u e^u\quad
&\mathrm{in}~\Omega,\\ u= u_0~&\mathrm{on}~\partial\Omega, \end{cases}
\end{equation} which corresponds to the stationary problem of a chemotaxis
system describing the aerobic bacterial movement, where $\Omega$ is a smooth
bounded domain in $\mathbb{R}^N (N\geq 1)$, $\varepsilon, m$ and $u_0$ are
positive constants. We show that the problem \eqref{h} admits a unique
classical solution which is of boundary-layer profile as $\varepsilon \to 0$,
where the boundary-layer thickness is of order $\varepsilon$. When
$\Omega=B_R(0)$ is a ball with radius $R>0$, we find a refined asymptotic
boundary layer profile up to the first-order expansion of $\varepsilon$ by
which we find that the slope of the layer profile in the immediate vicinity of
the boundary decreases with respect to (w.r.t.) the curvature while the
boundary-layer thickness increases {w.r.t.} the curvature.
