On the definition of the solution to a semilinear elliptic problem with a strong singularity at u=0. (December 2018)

• Record Type: Journal Article
• Authors: Giachetti, Daniela; Martínez-Aparicio, Pedro J.; Murat, François
• Is Part Of: Nonlinear analysis. Volume 177(2018)Part B
• Page Start: 491
• Page End: 523
• Publication Date: 2018-12
• Subjects: 35J25 -- 35J61 -- 35J75; Semilinear elliptic problems -- Singularity at u=0 -- Definition of the solution -- Zeroth order term with coefficient a measure -- Strong maximum principle; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.04.023
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we present new results related to the ones obtained in our previous papers on the singular semilinear elliptic problem u ≥ 0 in Ω, − d i v A ( x ) D u = F ( x, u ) in Ω, u = 0 on ∂ Ω, where F ( x, s ) is a Carathéodory function which can take the value + ∞ when s = 0 . Three new topics are investigated. First, we present definitions which we prove to be equivalent to the definition given in our paper Giachetti, Martínez-Aparicio, Murat (2018). Second, we study the set { x ∈ Ω : u ( x ) = 0 }, which is the set where the right-hand side of the equation could be singular in Ω, and we prove that actually, at almost every point x of this set, the right-hand side is non singular since one has F ( x, 0 ) = 0 . Third, we consider the case where a zeroth order term μ u, with μ a nonnegative bounded Radon measure which also belongs to H − 1 ( Ω ), is added to the left-hand side of the singular problem considered above. We explain how the definition of solution given in Giachetti, Martínez-Aparicio, Murat (2018) has to be modified in such a case, and we explicitly give the a priori estimates that every such solution satisfies (these estimates are at the basis of our existence, stability and uniqueness results). Finally we give two counterexamples which prove that when a zeroth order term μ u of the above type is added to the left-hand side of the problem, the strong maximum principle in general does not hold anymore.