Existence and comparison results for a doubly singular 1-Laplacian problem with $L^1$ data

Abstract

In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -\Delta_1 u + g(u)|Du| = h(u)f & \text{in }\Omega, u=0 & \text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(\Omega)$ is a nonnegative datum, and $g,h$ are nonnegative continuous functions on $(0,\infty)$ that may be singular at the origin. Under the minimal assumptions that $g$ is integrable near zero and $h$ is bounded at infinity, we explore the existence of a global $BV(\Omega)$ solution. Furthermore, a comparison principle is proved under suitable monotonicity assumptions on $h$. This framework avoids any growth restrictions on $h$ near the origin, thus allowing for highly singular terms. To handle these nonlinearities, we introduce a novel approach that takes advantage of the rigid structure of the 1-Laplacian operator.