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

TL;DR

This paper investigates the existence, uniqueness, and stability of solutions for a one-dimensional elliptic equation with a singular first-order divergence term, revealing unexpected phenomena and solution multiplicity.

Contribution

It provides a novel analysis of weak solutions for a singular elliptic PDE, including a detailed characterization and stability results, along with an associated ODE study.

Findings

Existence and uniqueness of solutions established.

A delicate stability property demonstrated.

Unexpected multiplicity of solutions discovered.

Abstract

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}.