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

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 ${\mathbb{R}}^2$, $x,y\in D$, $\hat{I}{›}_1,\hat{I}{›}_2>0$, ${\mathcal{R}}_D$ is the Robin function, and $G_D$ is the Green function of the operator $-\hat{I}”$ 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 ${\mathcal{K}\mathcal{R}}_D$, 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.

77 pages