Gelfand-type problems involving the 1-Laplacian operator
Molino, Alexis (Universidad de Almería. Departamento de Matemáticas.)
Segura de León, Sergio (Universitat de València. Departament d'Anàlisi Matemàtica)

Data:	2022
Resum:	In this paper, the theory of Gelfand problems is adapted to the 1-Laplacian setting. Concretely, we deal with the following problem: −∆1u = λf(u) in Ω,u = 0 on ∂Ω, where Ω ⊂ RN (N ≥ 1) is a domain, λ ≥ 0, and f : [0, +∞[ → ]0, +∞[ is any continuous increasing and unbounded function with f(0) > 0. We prove the existence of a threshold λ∗ = h(Ω) f(0) (h(Ω) being the Cheeger constant of Ω) such that there exists no solution when λ > λ∗ and the trivial function is always a solution when λ ≤ λ∗. The radial case is analyzed in more detail, showing the existence of multiple (even singular) solutions as well as the behavior of solutions to problems involving the p-Laplacian as p tends to 1, which allows us to identify proper solutions through an extra condition.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Nonlinear elliptic equations ; 1-Laplacian operator ; Gelfand problem
Publicat a:	Publicacions matemàtiques, Vol. 66 Núm. 1 (2022) , p. 269-304 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/396518
DOI: 10.5565/PUBLMAT6612211

36 p, 523.1 KB

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