Singular elliptic PDEs: an extensive overview

Francescantonio Oliva () and Francesco Petitta ()
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Francescantonio Oliva: “Sapienza” Università di Roma
Francesco Petitta: “Sapienza” Università di Roma

Partial Differential Equations and Applications, 2025, vol. 6, issue 1, 1-82

Abstract: 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 $$ -\Delta u = {u^{-\gamma }} \ {\text {in}}\ \Omega $$ - Δ u = u - γ in Ω where $$\Omega $$ Ω is a bounded subset of $$\mathbb {R}^N$$ R N ( $$N\ge 2$$ N ≥ 2 ), and $$\gamma >0$$ γ > 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.

Keywords: p-Laplacian; Nonlinear elliptic equations; Singular elliptic equations; Dirichlet boundary conditions; 35J25; 35J60; 35J75; 35R99; 35A02 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-024-00308-9

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