Unexpected phenomena in a one dimensional elliptic equation with a singular first order divergence term

In this paper we study the possible solutions $u$ of the one-dimensional non-linear singular problem which formally reads as \begin{equation}\label{intro}\tag{S} \begin{cases} \dys -\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 $L>0$, and where the data ($a$, $g$, $\phi$) are as follow: $a$ is a function of $L^\infty(0,L)$ which is bounded between two positive constants, $g$ is a function of $L^2(0,L)$, and the singular function $\phi:\mathbb{R}\mapsto \mathbb{R}\cup \{+\infty\}$ is continuous as a function with values in $\mathbb{R}\cup \{+\infty\}$, and satisfies $\phi(0)=+\infty$ and $\phi(s)<+\infty$ for every $s\in\mathbb{R}$, $s\not=0$; the model example for the singular function $\phi$ is $\phi_\gamma(s)={|s|^{-\gamma}}$ with $\gamma>0$. We first study the behaviour of the solutions of approximating problems (S$_n$) involving non-singular functions $\phi_n$ which converge to $\phi$ in a sense that we specify, and we prove that these solutions have subsequences which either converge to weak solutions of \eqref{intro} (for a definition of weak solutions that we specify), or converge to zero. We then prove that for a large class of data ($a$, $g$, $\phi$) it does not exist any weak solution of \eqref{intro}, while for another large class of data ($a$, $g$, $\phi$) it exists at least one weak solution of \eqref{intro}. Thanks to the study of an associated singular ODE (this study is of independent interest), we prove that under additional assumptions which are satisfied by the model example $\phi_\gamma(s)={|s|^{-\gamma}}$ when $0<\gamma<1$, if for some data ($a$, $g$, $\phi$) there exists one weak solution of \eqref{intro}, then for the same data it also exists an infinity number of weak solutions of \eqref{intro} which are parametrized by $c\in(-\infty,c^*]$ for some finite $c^*$. We finally prove that for any given data ($a$, $g$, $\phi$) and for any weak solution $u$ of \eqref{intro} corresponding to these data, there exist sequences of data ($a$, $g_n$, $\phi_n$), with non-singular functions $\phi_n$ which converge to ($a$, $g$, $\phi$), for which the solutions converge to $u$, while there also exist other sequences of data\break ($a$, $g_n$, $\phi_n$), with non-singular functions $\phi_n$ which converge to ($a$, $g$, $\phi$), for which the solutions converge to zero. Most of these results are unexpected.