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On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: Clustering concentration layers
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-08-26 , DOI: 10.1016/j.jfa.2021.109220
Suting Wei 1 , Jun Yang 2
Affiliation  

We consider the problemε2div(a(y)u)V(y)u+up=0,u>0in Ω,a(y)uν=0on Ω, where Ω is a bounded domain in R2 with smooth boundary, the exponent p is greater than 1, ε>0 is a small parameter, V is a uniformly positive smooth potential on Ω¯, and ν denotes the outward normal of ∂Ω. For two positive smooth functions a1(y),a2(y) on Ω¯, the operator a(y) is given bya(y)u=(a1(y)uy1,a2(y)uy2).

(1). Let ΓΩ¯ 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 R2 with metric V2σ(y)[a2(y)dy12+a1(y)dy22], where σ=p+1p112. By assuming some additional constraints on the functions a(y), V(y) and the curves Γ, ∂Ω, we prove that there exists a sequence of ε such that the problem has solutions uε with clustering concentration layers directed along Γ, exponentially small in ε at any positive distance from it.

(2). If Γ˜ 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 R2 with metric V2σ(y)[a2(y)dy12+a1(y)dy22], then a similar result of concentrated solutions is still true.



中文翻译:

关于二维光滑有界域的 Ambrosetti-Malchiodi-Ni 猜想:聚类浓度层

我们考虑问题ε2div(一种())-()+=0,>0在 Ω,一种()ν=0在 Ω, 其中 Ω 是一个有界域 电阻2边界平滑,指数p大于 1,ε>0是一个小参数,V是均匀正的平滑电位Ω¯, ν表示 ∂Ω 的外法线。对于两个正平滑函数一种1(),一种2()Ω¯, 运营商 一种() 是(谁)给的一种()=(一种1()1,一种2()2).

(1). 让ΓΩ¯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电阻2 带公制 2σ()[一种2()d12+一种1()d22], 在哪里 σ=+1-1-12. 通过对函数假设一些额外的约束一种(), ()和曲线 Γ, ∂Ω,我们证明存在一个ε序列,使得问题有解ε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Γ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电阻2 带公制 2σ()[一种2()d12+一种1()d22],那么浓缩溶液的类似结果仍然成立。

更新日期:2021-09-06
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