Qualitative analysis on the critical points of the Robin function

2022/02/22 by Francesca Gladiali, Massimo Grossi, Gladiali, Francesca +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2202.10895

Abstract

Let Ω⊂ℝ^N be a smooth bounded domain with N≥2 and Ω_ε=Ω\backslash B(P,ε) where B(P,ε) is the ball centered at P∈Ω 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 ∂ B(P,ε). Some applications to compute the exact number of solutions of related well-studied nonlinear elliptic problems will be showed.

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