Qualitative analysis on the critical points of the Robin function

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ be a smooth bounded domain with $N\ge 2$ and ${\Omega }_{\epsilon }=\Omega \setminus B(P,\epsilon )$, where $B(P,\epsilon )$ is the ball centered at $P\in \Omega$ with radius $\epsilon$. In this paper, we establish the number, location and non-degeneracy of critical points of the Robin function in ${\Omega }_{\epsilon }$ for $\epsilon$ small enough. We will show that the location of $P$ plays a crucial role in the existence and multiplicity of the critical points. The proof of our result is a consequence of delicate estimates on the Green’s function near $\partial B(P,\epsilon )$. Some applications to computing the exact number of solutions of related well-studied nonlinear elliptic problems are shown.