Bounded solutions to the 1-Laplacian equation with a total variation term

In this paper we study the Dirichlet problem for two related equations involving the 1-Laplacian and a total variation term as reaction, namely: with homogeneous Dirichlet boundary conditions on ∂ Ω , where Ω is a regular, bounded domain in R N . Here f is a measurable function belonging to some suitable Lebesgue space, while g ( u ) is a continuous function having the same sign as u and such that g ( ± ∞ ) = ± ∞ . As far as Eq. (1) is concerned, we show that a bounded solution exists if the datum f belongs to L N ( Ω ) . When the absorption term g ( u ) is missing, i.e. in the case of Eq. (2), we show that if f ∈ L N ( Ω ) , and its norm is small, then the only solution of (2) is u ≡ 0 . In the case where the norm of f is not small, several cases may happen. Depending on Ω and f , we show examples where no solution of (2) exists, other examples where u ≡ 0 is still a solution, and finally examples with nontrivial solutions. Some of these results can be viewed as a translation to the 1-Laplacian operator of known results by Ferone and Murat.