Qualitative analysis on the critical points of the Robin function
Francesca Gladiali, Massimo Grossi, Peng Luo, Shusen Yan · Journal of the European Mathematical Society · 2024
Let \Omega\subset\mathbb{R}^{N} be a smooth bounded domain with N\ge2 and \Omega_{\varepsilon}=\Omega\setminus B(P,\varepsilon) , where B(P,\varepsilon) is the ball centered at P\in\Omega with radius \varepsilon . In this paper, we establish the number, location and non-degeneracy of critical points of the Robin function in \Omega_{\varepsilon} for \varepsilon 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,\varepsilon) . Some applications to computing the exact number of solutions of related well-studied nonlinear elliptic problems are shown.