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} -\Delta_1u=\lambda f(u) &\hbox{in }\Omega\,;\\[2mm] u=0 &\hbox{on }\partial\Omega\,; \end{array} \right. \end{equation*} where \(\Omega\subset\mathbb{R}^N\) (\(N\ge1\)) is a domain, \(\lambda \geq 0\) 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 \(\lambda^*=\frac{h(\Omega)}{f(0)}\) (being \(h(\Omega)\) the Cheeger constant of \(\Omega\)) such that there exists no solution when \(\lambda>\lambda^*\) and the trivial function is always a solution when \(\lambda\le\lambda^*\). 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.