Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0

2016/06/23 by Daniela Giachetti, Giachetti, Daniela, Pedro J. Martínez−Aparicio +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1606.07267

openalex publication_date 2016/06/23 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper we consider a semilinear elliptic equation with a strong singularity at u=0, namely u≥ 0 in Ω, - div A(x) D u = F(x,u) in Ω, u = 0 on ∂ Ω, with F(x,s) a Carathéodory function such that 0≤ F(x,s)≤ (h(x))/(Γ(s)) a.e. x∈Ω, ∀ sgt;0, with h in some L^r(Ω) and Γ a C¹([0,+∞[) function such that Γ(0)=0 and Γ'(s)>0 for every s>0. We introduce a notion of solution to this problem in the spirit of the solutions defined by transposition. This definition allows us to prove the existence and the stability of this solution, as well as its uniqueness when F(x,s) is nonincreasing in s.

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