The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions

<p>The purpose of this paper is to study and classify singular solutions of the Poisson problem <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row script upper L Subscript mu Superscript s Baseline u equals f in normal upper Omega minus StartSet 0 EndSet comma 2nd Row u equals 0 in double-struck upper R Superscript upper N Baseline minus normal upper Omega EndLayout"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>s</mml:mi> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mspace width="1em" /> <mml:mtext> </mml:mtext> <mml:mtext>in</mml:mtext> <mml:mtext> </mml:mtext> <mml:mspace width="thinmathspace" /> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="1em" /> <mml:mtext> </mml:mtext> <mml:mtext>in</mml:mtext> <mml:mtext> </mml:mtext> <mml:mspace width="thinmathspace" /> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mtext> </mml:mtext> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence="true" stretchy="true" symmetric="true" /> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \left \{ \begin {aligned} \mathcal {L}^s_\mu u = f \quad \ \text {in}\ \, \Omega \setminus \{0\},\\ u =0 \quad \ \text {in}\ \, \mathbb {R}^N \setminus \Omega \ \end{aligned} \right . \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> for the fractional Hardy operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript mu Superscript s Baseline u equals left-parenthesis negative normal upper Delta right-parenthesis Superscript s Baseline u plus StartFraction mu Over StartAbsoluteValue x EndAbsoluteValue Superscript 2 s Baseline EndFraction u"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>s</mml:mi> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>s</mml:mi> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mi>μ<!-- μ --></mml:mi> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}_\mu ^s u= (-\Delta )^s u +\frac {\mu }{|x|^{2s}}u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a bounded domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of double-struck upper R Superscript upper N"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \subset \mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N \ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) containing the origin. Here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative normal upper Delta right-parenthesis Superscript s"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(-\Delta )^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s element-of left-parenthesis 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">s\in (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is the fractional Laplacian of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 s"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu greater-than-or-equal-to mu 0"> <mml:semantics> <mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu \ge \mu _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu 0 equals minus 2 Superscript 2 s Baseline StartStartFraction normal upper Gamma squared left-parenthesis StartFraction upper N plus 2 s Over 4 EndFraction right-parenthesis OverOver normal upper Gamma squared left-parenthesis StartFraction upper N minus 2 s Over 4 EndFraction right-parenthesis EndEndFraction greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu _0 = -2^{2s}\frac {\Gamma ^2(\frac {N+2s}4)}{\Gamma ^2(\frac {N-2s}{4})}>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the b

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