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.
2025
$p$-Laplacian, Nonlinear elliptic equations, Singular elliptic equations, Dirichlet boundary conditions
01 Pubblicazione su rivista::01a Articolo in rivista
Singular Elliptic PDEs: an extensive overview / Oliva, Francescantonio; Petitta, Francesco. - In: SN PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 2662-2963. - (2025).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1731448
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