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: egin{equation}label{e0} g( u)-hbox{div,}Big(rac{Du}{|Du|}Big)=|Du|+f(x),, end{equation} egin{equation}label{e1} -hbox{div,}Big(rac{Du}{|Du|}Big)=|Du|+f(x),, end{equation} with homogeneous Dirichlet boundary conditions on $partial Omega$, where $Omega$ is a regular, bounded domain in $mathbb{R}^N$. 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 $g(pm infty) = pminfty$. As far as equation ( ef{e0}) is concerned, we show that a bounded solution exists if the datum $f$ belongs to $L^N(Omega)$. When the absorption term $g(u)$ is missing, i.e. in the case of equation ( ef{e1}), we show that if $fin L^N(Omega)$, and its norm is small, then the only solution of ( ef{e1}) is $uequiv0$. In the case where the norm of $f$ is not small, several cases may happen. Depending on $Omega$ and $f$, we show examples where no solution of ( ef{e1}) exists, other examples where $uequiv0$ 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.