Qualitative analysis on the critical points of the Kirchhoff-Routh function
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911343031877632 |
|---|---|
| author | Gladiali, Francesca Grossi, Massimo Luo, Peng Yan, Shusen |
| author_facet | Gladiali, Francesca Grossi, Massimo Luo, Peng Yan, Shusen |
| contents | In this paper, we study the number of critical points of the Kirchhoff-Routh function \begin{equation*} \mathcal{KR}_D(x,y)=Λ_1^2\mathcal{R}_D(x)+Λ_2^2\mathcal{R}_D(y)-2Λ_1Λ_2G_D(x,y), \end{equation*} where $D$ is a bounded domain in $\mathbb{R}^2$, $x,y\in D$, $Λ_1,Λ_2>0$, $\mathcal{R}_D$ is the Robin function, and $G_D$ is the Green function of the operator $-Δ$ with $0$ Dirichlet boundary condition on $D$. This function arises from concentration phenomena in nonlinear elliptic problems and from the de-singularization problem for the steady Euler equation. For domains with a small hole, we establish not only the exact number and the location of the critical points of $\mathcal{KR}_D$, but also their nondegeneracy. We show that the location of the hole plays a crucial role. Finally in the context of elliptic problems, we establish the existence of multiple two-peak solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23172 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Qualitative analysis on the critical points of the Kirchhoff-Routh function Gladiali, Francesca Grossi, Massimo Luo, Peng Yan, Shusen Analysis of PDEs 35A02, 35J08, 35J60 In this paper, we study the number of critical points of the Kirchhoff-Routh function \begin{equation*} \mathcal{KR}_D(x,y)=Λ_1^2\mathcal{R}_D(x)+Λ_2^2\mathcal{R}_D(y)-2Λ_1Λ_2G_D(x,y), \end{equation*} where $D$ is a bounded domain in $\mathbb{R}^2$, $x,y\in D$, $Λ_1,Λ_2>0$, $\mathcal{R}_D$ is the Robin function, and $G_D$ is the Green function of the operator $-Δ$ with $0$ Dirichlet boundary condition on $D$. This function arises from concentration phenomena in nonlinear elliptic problems and from the de-singularization problem for the steady Euler equation. For domains with a small hole, we establish not only the exact number and the location of the critical points of $\mathcal{KR}_D$, but also their nondegeneracy. We show that the location of the hole plays a crucial role. Finally in the context of elliptic problems, we establish the existence of multiple two-peak solutions. |
| title | Qualitative analysis on the critical points of the Kirchhoff-Routh function |
| topic | Analysis of PDEs 35A02, 35J08, 35J60 |
| url | https://arxiv.org/abs/2512.23172 |