We consider the problem${\epsilon }^2\mathrm{div}\left({\mathrm{\nabla }}_{\mathfrak{a}(y)}u\right)-V(y)u+u^p=0,u>0\text{in }\mathrm{\Omega },$${\mathrm{\nabla }}_{\mathfrak{a}(y)}u\cdot \nu =0\text{on }\partial \mathrm{\Omega },$ where Ω is a bounded domain in ${\mathbb{R}}^2$ with smooth boundary, the exponent p is greater than 1, $\epsilon >0$ is a small parameter, V is a uniformly positive smooth potential on $\overline{\mathrm{\Omega }}$, and ν denotes the outward normal of ∂Ω. For two positive smooth functions ${\mathfrak{a}}_1(y),{\mathfrak{a}}_2(y)$ on $\overline{\mathrm{\Omega }}$, the operator ${\mathrm{\nabla }}_{\mathfrak{a}(y)}$ is given by${\mathrm{\nabla }}_{\mathfrak{a}(y)}u=({\mathfrak{a}}_1(y)\frac{\partial u}{\partial y_1},{\mathfrak{a}}_2(y)\frac{\partial u}{\partial y_2}).$

(1). Let $\mathrm{\Gamma }\subset \overline{\mathrm{\Omega }}$ be a smooth curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two parts. Moreover, Γ is a non-degenerate geodesic embedded in the Riemannian manifold ${\mathbb{R}}^2$ with metric $V^{2\sigma }(y)[{\mathfrak{a}}_2(y)\mathrm{d}{y}_1^2+{\mathfrak{a}}_1(y)\mathrm{d}{y}_2^2]$, where $\sigma =\frac{p+1}{p-1}-\frac{1}{2}$. By assuming some additional constraints on the functions $\mathfrak{a}(y)$, $V(y)$ and the curves Γ, ∂Ω, we prove that there exists a sequence of ε such that the problem has solutions $u_{\epsilon }$ with clustering concentration layers directed along Γ, exponentially small in ε at any positive distance from it.

(2). If $\tilde{\mathrm{\Gamma }}$ is a simple closed smooth curve in Ω (not touching the boundary ∂Ω), which is also a non-degenerate geodesic embedded in the Riemannian manifold ${\mathbb{R}}^2$ with metric $V^{2\sigma }(y)[{\mathfrak{a}}_2(y)\mathrm{d}{y}_1^2+{\mathfrak{a}}_1(y)\mathrm{d}{y}_2^2]$, then a similar result of concentrated solutions is still true.

我们考虑问题${\epsilon }^2\mathrm{div}\left({\mathrm{\nabla }}_{\mathfrak{一种}(是)}你\right)-伏(是)你+{你}^{磷}=0,你>0\text{在 }\mathrm{\Omega },$${\mathrm{\nabla }}_{\mathfrak{一种}(是)}你\cdot \nu =0\text{在 }\partial \mathrm{\Omega },$ 其中 Ω 是一个有界域 ${\mathbb{电阻}}^2$边界平滑，指数p大于 1，$\epsilon >0$是一个小参数，V是均匀正的平滑电位$\overline{\mathrm{\Omega }}$, ν表示 ∂Ω 的外法线。对于两个正平滑函数${\mathfrak{一种}}_1(是),{\mathfrak{一种}}_2(是)$ 在 $\overline{\mathrm{\Omega }}$， 运营商 ${\mathrm{\nabla }}_{\mathfrak{一种}(是)}$ 是（谁）给的${\mathrm{\nabla }}_{\mathfrak{一种}(是)}你=({\mathfrak{一种}}_1(是)\frac{\partial 你}{\partial {是}_1},{\mathfrak{一种}}_2(是)\frac{\partial 你}{\partial {是}_2}).$

(1). 让$\mathrm{\Gamma }\subset \overline{\mathrm{\Omega }}$是一条平滑的曲线，与 ∂Ω 正交，正好在两点处，并将 Ω 分成两部分。此外，Γ 是嵌入黎曼流形的非退化测地线${\mathbb{电阻}}^2$ 带公制 ${伏}^{2\sigma }(是)[{\mathfrak{一种}}_2(是)\mathrm{d}{是}_1^2+{\mathfrak{一种}}_1(是)\mathrm{d}{是}_2^2]$， 在哪里 $\sigma =\frac{磷+1}{磷-1}-\frac{1}{2}$. 通过对函数假设一些额外的约束$\mathfrak{一种}(是)$, $伏(是)$和曲线 Γ, ∂Ω，我们证明存在一个ε序列，使得问题有解${你}_{\epsilon }$具有沿 Γ 指向的聚类浓度层，在与它的任何正距离处，ε呈指数级小。

(2). 如果$\stackrel{～}{\mathrm{\Gamma }}$是 Ω 中的简单闭合平滑曲线（不接触边界 ∂Ω），它也是嵌入黎曼流形中的非退化测地线${\mathbb{电阻}}^2$ 带公制 ${伏}^{2\sigma }(是)[{\mathfrak{一种}}_2(是)\mathrm{d}{是}_1^2+{\mathfrak{一种}}_1(是)\mathrm{d}{是}_2^2]$，那么浓缩溶液的类似结果仍然成立。