Gelfand-type problems involving the 1-Laplacian operator

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.