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

2025/12/29 by Gladiali, Francesca, Grossi, Massimo, Luo, Peng +1
#35A02 #35J08 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.23172

Abstract

In this paper, we study the number of critical points of the Kirchhoff-Routh function KR_D(x,y)=Λ₁²R_D(x)+Λ₂²R_D(y)-2Λ₁Λ₂G_D(x,y), where D is a bounded domain in ℝ², x,y∈ D, Λ₁,Λ₂>0, 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 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.

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