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 \begin{equation*} \left\{\begin{array}{cc} -Δ_1u=λf(u) &\hbox{in }Ω\,;\\[2mm] u=0 &\hbox{on }\partialΩ\,; \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$ ($N\ge1$) is a domain, $λ\geq 0$ and $f\>:\>[0,+\infty[\to]0,+\infty[$ is any continuous increasing and unbounded function with $f(0)>0$. It is proved the existence of a threshold $λ^*=\frac{h(Ω)}{f(0)}$ (being $h(Ω)$ the Cheeger constant of $Ω$) such that there exists no solution when $λ>λ^*$ and the trivial function is always a solution when $λ\leλ^*$. The radial case is analyzed in more detail showing the existence of multiple solutions (even singular) as well as the behaviour 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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