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Qualitative analysis on the critical points of the Robin function

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