  We study existence of a weak solution for one-dimensional problems as
\begin{equation}\label{intro}\tag{1} \begin{cases} \displaystyle
-\frac{d}{dx}\left(a(x) \frac{d u}{dx}\right) = - \frac{d \phi (u) }{dx}-
\frac{d g(x) }{dx}& \text{in}\;(0,L),\\
  u(0)=u(L)=0\,, & \end{cases} \end{equation} where $a$ is a positive bounded
function, $g\in L^2(0,L)$, and $\phi:\mathbb{R}\mapsto \mathbb{R}\cup
\{+\infty\}$ is continuous as a function with values in $\mathbb{R}\cup
\{+\infty\}$. Some relevant qualitative and quantitative facts concerning such
problems and their solutions are described. In particular a precise
characterization of the behaviour of suitable approximating solution is
provided. Of particular (and independent) interest is the study of an
associated ODE for which, we prove existence, uniqueness and comparison
results. As a consequence of our arguments, a delicate stability result as well
a quite unexpected multiplicity result is shown for problems as in
\eqref{intro}.
