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 $\gamma>0$. \smallskip 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.
Singular Elliptic PDEs: an extensive overview / Oliva, Francescantonio; Petitta, Francesco. - In: SN PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 2662-2963. - (2025).
Singular Elliptic PDEs: an extensive overview
Francescantonio Oliva;Francesco Petitta
2025
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 $\gamma>0$. \smallskip 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.