Singular Elliptic PDEs: an extensive overview

TL;DR

This survey provides a comprehensive overview of nonlinear elliptic PDEs with singular zero-order terms, discussing existence, uniqueness, regularity, and open problems in the field.

Contribution

It offers a modern, self-contained overview of key results and open problems in singular elliptic PDEs, including both classical and recent advances.

Findings

Summary of existence and uniqueness results

Discussion on regularity depending on data

Presentation of open problems in the field

Abstract

In this survey we provide an overview of nonlinear elliptic homogeneous boundary value problems featuring singular zero-order terms with respect to the unknown variable whose prototype equation is $$ -\Delta u = {u^{-\gamma}} \ \text{in}\ \Omega $$ where $\Omega$ is a bounded subset of $\mathbb{R}^N$ ($N\geq 2$), and $\gamma>0$. We start by outlining the basic concepts and the mathematical framework needed for setting the problem. Both old and new key existence and uniqueness results are presented, alongside regularity issues depending on the regularity of the data. The presentation aims to be modern, self-contained and consistent. Some examples and open problems are also discussed.