New definitions of a solution of a one-dimensional elliptic equation with a singular first order divergence term

In this paper, which is a continuation of our recent paper [1], we give two new definitions of a weak solution of the one-dimensional, elliptic, nonlinear singular problem which is formally written as \begin{equation*} \begin{cases} -\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 $\phi:\mathbb{R}\mapsto\mathbb{R}\cup\{+\infty\}$ is a singular function whose model is given by $\phi(s)=\phi_\gamma (s)=\frac{1}{|s|^{\gamma}}$ with $\gamma>0$. In these two definitions the solutions belong respectively to the space $u\in W_0^{1,1}(0,L)$ with $\phi(u)\in L^1(0,L)$, and to the space $u\in W_0^{1,m}(0,L)$ with $\phi(u)\in L^m(0,L)$ for $m>1$. In these frameworks we state and prove results of non-existence, existence, and non-isolation of the solutions.