The Dirichlet problem for singular elliptic equations with general nonlinearities

TL;DR

This paper establishes existence, regularity, and uniqueness of solutions for a class of singular elliptic equations involving the 1-Laplace operator with general nonlinearities, extending the theory to include p-Laplacian cases.

Contribution

It introduces a general framework for solving singular elliptic equations with the 1-Laplace operator and extends results to p-Laplacian cases, filling gaps in existing literature.

Findings

Proved existence and regularity of solutions under broad assumptions.

Established uniqueness when the nonlinearity is decreasing and data is positive.

Developed a general theory applicable to p-Laplacian problems.

Abstract

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline u\geq 0& \text{in}\ \Omega, \newline u=0 & \text{on}\ \partial \Omega, \end{cases} $$ where, $\Delta_{1} $ is the $1$-laplace operator, $\Omega$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $h(s)$ is a continuous function which may become singular at $s=0^{+}$, and $f$ is a nonnegative datum in $L^{N,\infty}(\Omega)$ with suitable small norm. Uniqueness of solutions is also shown provided $h$ is decreasing and $f>0$. As a by-product of our method a general theory for the same problem involving the $p$-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality.