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)
Date: |
2022 |
Abstract: |
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. |
Rights: |
Tots els drets reservats. |
Language: |
Anglès |
Document: |
Article ; recerca ; Versió publicada |
Subject: |
Nonlinear elliptic equations ;
1-Laplacian operator ;
Gelfand problem |
Published in: |
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
The record appears in these collections:
Articles >
Published articles >
Publicacions matemàtiquesArticles >
Research articles
Record created 2022-02-03, last modified 2022-09-03