Title: Singular Elliptic PDEs: an extensive overview Show Chinese title

Title: 奇异椭圆PDEs：全面概述

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. Show Chinese abstract

Abstract: 在本次综述中，我们提供了一个非线性椭圆齐次边界值问题的概述，这些问题具有关于未知变量的奇异零阶项，其原型方程是$$ -\Delta u = {u^{-\gamma}} \ \text{in}\ \Omega $$，其中$\Omega$是$\mathbb{R}^N$($N\geq 2$) 中的一个有界子集，以及$\gamma>0$。 我们首先概述了设定该问题所需的基本概念和数学框架。 介绍了旧的和新的关键存在性和唯一性结果，并讨论了依赖于数据正则性的正则性问题。 此介绍旨在现代、自包含且一致。 还讨论了一些例子和开放问题。