We study the local properties of positive solutions of the equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3118_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">$-\Delta u+ae^{bu}=m\left| \nabla u\right| ^q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow> <mi mathvariant="italic">bu</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <msup> <mfenced close="|" open="|"> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mfenced> <mi>q</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> in a punctured domain <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3118_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \setminus \{0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3118_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> where m, a, b are positive parameters and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3118_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$q>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We study in particular the existence of solutions with an isolated singularity and the local behaviour of such singular solutions.

Isolated singularities of solutions of a 2-D diffusion equation with mixed reaction

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发表日期：

2025年8月31日

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Yimei Li,
Laurent Véron

摘要

We study the local properties of positive solutions of the equation $-\Delta u+ae^{bu}=m\left| \nabla u\right| ^q$ $- Δ u + a e^{bu} = m {|\nabla, u|}^{q}$ in a punctured domain $\Omega \setminus \{0\}$ $Ω \ {0}$ of ${\mathbb {R}}^2$ $R^{2}$ where m, a, b are positive parameters and $q>1$ $q > 1$ . We study in particular the existence of solutions with an isolated singularity and the local behaviour of such singular solutions.