On the number and location of critical points of solutions of nonlinear elliptic equations in domains with a small hole

2020/03/07 by Massimo Grossi, Peng Luo, Grossi, Massimo +1
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2003.03643

openalex publication_date 2020/03/07 · openalex created_date 2020/03/13 · openalex updated_date 2026/07/28

Abstract

In this paper we study the following problem \begincases -Δu=f(u)~amp;in Ω_ε,
ugt;0~amp;in Ω_ε,
u=0~amp;on ∂Ω_ε, \endcases where Ω_ε=Ω\backslash B(P,ε), Ω⊂ R^N with N≥ 2 is a smooth bounded domain, B(P,ε) is the ball centered at P and radius ε>0 and f is a smooth nonlinearity. By some computations involving the Green function and degree theory, we compute the number and location of critical points of solutions for small ε>0.

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