Gelfand-type problems involving the 1-Laplacian operator

  1. Molino, Alexis
  2. Segura de León, Sergio
Revista:
Publicacions matematiques

ISSN: 0214-1493

Año de publicación: 2022

Volumen: 66

Número: 1

Páginas: 269-304

Tipo: Artículo

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Otras publicaciones en: Publicacions matematiques

Resumen

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.

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