LAUSR

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 $$\partial \varOmega $$∂Ω, where $$\varOmega $$Ω is a regular, bounded domain in $$\mathbb {R}^N$$RN. 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 $$g(\pm \infty ) = \pm \infty $$g(±∞)=±∞. As far as Eq. (1) is concerned, we show that a bounded solution exists if the datum f belongs to $$L^N(\varOmega )$$LN(Ω). When the absorption term g(u) is missing, i.e. in the case of Eq. (2), we show that if $$f\in L^N(\varOmega )$$f∈LN(Ω), and its norm is small, then the only solution of (2) is $$u\equiv 0$$u≡0. In the case where the norm of f is not small, several cases may happen. Depending on $$\varOmega $$Ω and f, we show examples where no solution of (2) exists, other examples where $$u\equiv 0$$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.