Qualitative analysis on the critical points of the Robin function

TL;DR

This paper investigates how the critical points of the Robin function behave in a domain with a small removed ball, revealing their number, location, and stability, with implications for nonlinear elliptic equations.

Contribution

It provides new insights into the critical points of the Robin function in perturbed domains, especially regarding their existence, multiplicity, and non-degeneracy for small perturbations.

Findings

Number and location of critical points depend on the point P.

Critical points are non-degenerate under certain conditions.

Results help determine solutions to related nonlinear elliptic problems.

Abstract

Let $\Omega\subset\mathbb{R}^N$ be a smooth bounded domain with $N\ge2$ and $\Omega_\epsilon=\Omega\backslash B(P,\epsilon)$ where $B(P,\epsilon)$ is the ball centered at $P\in\Omega$ and 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 on the existence and multiplicity of the critical points. The proof of our result is a consequence of delicate estimates on the Green function near to $\partial B(P,\epsilon)$. Some applications to compute the exact number of solutions of related well-studied nonlinear elliptic problems will be showed.