  We consider the partial differential equation $$ u-f={\rm
div}\left(u^m\frac{\nabla u}{|\nabla u|}\right) $$ with $f$ nonnegative and
bounded and $m\in\mathbb{R}$. We prove existence and uniqueness of solutions
for both the Dirichlet problem (with bounded and nonnegative {boundary datum})
and the homogeneous Neumann problem. Solutions, which a priori belong to a
space of truncated bounded variation functions, are shown to have zero jump
part with respect to the ${\mathcal H}^{N-1}$ Haussdorff measure. Results and
proofs extend to more general nonlinearities.
