Qualitative analysis on the critical points of the Robin function

Let $\O\subset\R^N$ be a smooth bounded domain with $N\ge2$ and $\O_\e=\O\backslash B(P,\e)$ where $B(P,\e)$ is the ball centered at $P\in\O$ and radius $\e$. In this paper, we establish the number, location and non-degeneracy of critical points of the Robin function in $\O_\e$ for $\e$ 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 $\partial B(P,\e)$. Some applications to compute the exact number of solutions of related well-studied nonlinear elliptic problems will be showed.