In this paper, the theory of Gelfand problems is adapted to the 1-Laplacian setting. Concretely, we deal with the following problem:
where () is a domain, , and is any continuous increasing and unbounded function with .
We prove the existence of a threshold ( being the Cheeger constant of ) such that there exists no solution when and the trivial function is always a solution when . 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 -Laplacian as tends to , which allows us to identify proper solutions through an extra condition.
"Gelfand-type problems involving the 1-Laplacian operator." Publ. Mat. 66 (1) 269 - 304, 2022. https://doi.org/10.5565/PUBLMAT6612211