Qualitative analysis on the critical points of the Robin function

Let ΩN be a smooth bounded domain with N≥2 and Ωε=Ω B(P,ε) , where B(P,ε) is the ball centered at P∈Ω with radius ε . In this paper, we establish the number, location and non-degeneracy of critical points of the Robin function in Ωε for ε 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 ∂ B(P,ε) . Some applications to computing the exact number of solutions of related well-studied nonlinear elliptic problems are shown.