<p>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 <Equation ID="Equ199"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2024_308_Article_Equ199.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </MediaObject> <EquationSource Format="TEX">$ -\Delta u = {u^{-\gamma }} \ {\text {in}}\ \Omega $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <msup> <mi>u</mi> <mrow> <mo>-</mo> <mi>γ</mi> </mrow> </msup> <mspace width="4pt" /> <mi mathvariant="normal">in</mi> <mspace width="4pt" /> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2024_308_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a bounded subset of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2024_308_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2024_308_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$N\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>), and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2024_308_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

Singular elliptic PDEs: an extensive overview

期刊论文
发表日期：

2025年2月5日

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Francescantonio Oliva,
Francesco Petitta

摘要

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 $\mathbb {R}^N$ $R^{N}$ ( $N\ge 2$ $N \geq 2$ ), 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.