Qualitative analysis on the critical points of the Robin function

Abstract

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