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.

中文翻译：

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趋化性引起的奇异扰动非局部半线性问题的边界层剖面

本文关注以下奇异摄动非局部半线性问题 \begin{equation} \label{h} \tag{$\ast$} \begin{cases} \varepsilon^2 \Delta u=\frac{ m}{\int_{\Omega}e^{u}{\mathrm{d}x}}ue^u\quad &\mathrm{in}~\Omega,\\ u= u_0~&\mathrm{on} ~\partial\Omega, \end{cases} \end{equation} 对应于描述有氧细菌运动的趋化系统的平稳问题，其中 $\Omega$ 是 $\mathbb{R}^ 中的平滑有界域N (N\geq 1)$、$\varepsilon、m$ 和 $u_0$ 是正常数。我们表明问题 \eqref{h} 承认一个独特的经典解，其边界层剖面为 $\varepsilon \to 0$，其中边界层厚度为 $\varepsilon$。当$\Omega=B_R(0)$为半径$R>0$的球时，