Gelfand-type problems involving the 1-Laplacian operator

Abstract

In this paper, the theory of Gelfand problems is adapted to the 1-Laplacian setting. Concretely, we deal with the following problem:

where $\mathrm{\Omega }\subset {ℝ}^N$ ($N\ge 1$) is a domain, $\lambda \ge 0$, and $f:[0,+\infty [\to ]0,+\infty [$ is any continuous increasing and unbounded function with .

We prove the existence of a threshold ${\lambda }^{*}=\frac{h(\mathrm{\Omega })}{f(0)}$ ($h(\mathrm{\Omega })$ being the Cheeger constant of $\mathrm{\Omega }$) such that there exists no solution when 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 (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.