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Qualitative analysis on the critical points of the Kirchhoff-Routh function

Francesca Gladiali, Massimo Grossi, Peng Luo, Shusen Yan

TL;DR

This work analyzes the critical points of the Kirchhoff-Routh function $\mathcal{KR}_{2,D}$ in planar domains with a small hole, revealing a rich bifurcation structure into three types (I–III) of concentration patterns. By deriving precise asymptotic expansions and employing degree theory, the authors establish existence, multiplicity, and nondegeneracy results for each type, highlighting the hole location and domain geometry as decisive factors. In particular, type I points persist as perturbations of the unperturbed problem; type II points occur under delicate balance near the hole with strong dependence on the domain and parameters; and type III points always exist and, under generic conditions (e.g., $\Lambda_1\neq\Lambda_2$ and $\nabla\mathcal{R}_{\Omega}(P)\neq0$), yield exactly two nondegenerate critical points with explicit asymptotics. These structural results translate into the existence of two-peak solutions for related nonlinear elliptic problems (Gel’fand, Lane-Emden) and their desingularization in 2D, offering a detailed map from vortex-like critical points of $\mathcal{KR}$ to concentration phenomena in PDEs.

Abstract

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.

Qualitative analysis on the critical points of the Kirchhoff-Routh function

TL;DR

This work analyzes the critical points of the Kirchhoff-Routh function in planar domains with a small hole, revealing a rich bifurcation structure into three types (I–III) of concentration patterns. By deriving precise asymptotic expansions and employing degree theory, the authors establish existence, multiplicity, and nondegeneracy results for each type, highlighting the hole location and domain geometry as decisive factors. In particular, type I points persist as perturbations of the unperturbed problem; type II points occur under delicate balance near the hole with strong dependence on the domain and parameters; and type III points always exist and, under generic conditions (e.g., and ), yield exactly two nondegenerate critical points with explicit asymptotics. These structural results translate into the existence of two-peak solutions for related nonlinear elliptic problems (Gel’fand, Lane-Emden) and their desingularization in 2D, offering a detailed map from vortex-like critical points of to concentration phenomena in PDEs.

Abstract

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 is a bounded domain in , , , is the Robin function, and is the Green function of the operator with Dirichlet boundary condition on . 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 , 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.
Paper Structure (23 sections, 71 theorems, 576 equations, 3 figures)

This paper contains 23 sections, 71 theorems, 576 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Critical points of type I
  3. Critical points of type II
  4. Critical points of type III
  5. Summary and examples
  6. Applications to nonlinear elliptic problems
  7. Outlines of the proofs of the main results
  8. A first necessary condition of critical points
  9. The critical points of type I
  10. The critical points of type II
  11. The existence of critical points of Type III
  12. The location of critical points
  13. Existence and asymptotics
  14. The exact multiplicity of type III critical points
  15. The improved expansion for $\nabla\mathcal{KR}_{\Omega_\varepsilon}(x,y)$
  16. ...and 8 more sections

Key Result

Proposition 1.2

Let $(x_\varepsilon,y_\varepsilon)$ be a critical point of $\mathcal{KR}_{\Omega_\varepsilon}(x,y)$ with $(x_\varepsilon,y_\varepsilon)\to (x_0,y_0)\in \overline \Omega\times \overline \Omega$ as $\varepsilon\to 0$. Then (1) there exists a positive constant $\delta$ such that (2) if $x_0=y_0$, then it holds $x_0=y_0=P$.

Figures (3)

  • Figure 1: The case where $(x_\varepsilon,y_\varepsilon)\to (x_0,y_0)$ with $x_0,y_0\ne P$
  • Figure 2:
  • Figure :

Theorems & Definitions (159)

  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 149 more