<p>In this paper, we use the Caffarelli–Silvestre extension to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb R_+\times \mathbb R^N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo>×</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> to study the isolated singularities of functions satisfying the semilinear fractional equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(-\Delta )^sv+\epsilon v^p=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> <mi>v</mi> <mo>+</mo> <mi>ϵ</mi> <msup> <mi>v</mi> <mi>p</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in a punctured domain of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb R^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\epsilon =\pm 1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$0<s<1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p>1.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We emphasise the derivation of a priori estimates and analyse the set of self-similar solutions. We provide a complete description of the possible behavior of solutions near a singularity.</p>

APPLICATIONS OF THE s-HARMONIC EXTENSION TO THE STUDY OF SINGULARITIES OF EMDEN’S EQUATIONS

期刊论文
发表日期：

2025年9月1日

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L. Véron

摘要

In this paper, we use the Caffarelli–Silvestre extension to $\mathbb R_+\times \mathbb R^N$ $R_{+} \times R^{N}$ to study the isolated singularities of functions satisfying the semilinear fractional equation $(-\Delta )^sv+\epsilon v^p=0$ ${(- Δ)}^{s} v + ϵ v^{p} = 0$ in a punctured domain of $\mathbb R^N$ $R^{N}$ with $\epsilon =\pm 1,$ $ϵ = \pm 1,$ $0<s<1,$ $0 < s < 1,$ and $p>1.$ $p > 1 .$ We emphasise the derivation of a priori estimates and analyse the set of self-similar solutions. We provide a complete description of the possible behavior of solutions near a singularity.