Multiplicity and Stability of the Pohozaev Obstruction for Hardy-Schrödinger Equations with Boundary Singularity
Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth bounded domain in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <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:annotation encoding="application/x-tex">\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="n greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 element-of partial-differential normal upper Omega"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>∈</mml:mo> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0\in \partial \Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We consider issues of non-existence, existence, and multiplicity of variational solutions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript 1 comma 0 Superscript 2 Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_{1,0}^2(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the borderline Dirichlet 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 1st Column minus normal upper Delta u minus gamma StartFraction u Over StartAbsoluteValue x EndAbsoluteValue squared EndFraction minus h left-parenthesis x right-parenthesis u 2nd Column a m p semicolon equals 3rd Column a m p semicolon StartFraction StartAbsoluteValue u EndAbsoluteValue Superscript 2 Super Superscript star Superscript left-parenthesis s right-parenthesis minus 2 Baseline u Over StartAbsoluteValue x EndAbsoluteValue Superscript s Baseline EndFraction 4th Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u 2nd Column a m p semicolon equals 3rd Column a m p semicolon 0 4th Column a m p semicolon on partial-differential normal upper Omega minus StartSet 0 EndSet comma EndLayout"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left left left left" rowspacing="4pt" columnspacing="1em"> <mml:mtr> <mml:mtd> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mi>γ</mml:mi> <mml:mfrac> <mml:mi>u</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:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>u</mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mfrac> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>u</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:msup> <mml:mn>2</mml:mn> <mml:mo>⋆</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> </mml:mrow> <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:mi>s</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mtext> </mml:mtext> <mml:mtext> </mml:mtext> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext>in </mml:mtext> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd columnalign="right"> <mml:mi>u</mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mtext>on </mml:mtext> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi mathvariant="normal">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:mtable> <mml:mo fence="true" stretchy="true" symmetric="true"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \left \{ \begin {array}{llll} -\Delta u-\gamma \frac {u}{|x|^2}- h(x) u &=& \frac {|u|^{2^\star (s)-2}u}{|x|^s} \ \ &\text {in } \Omega ,\\ \hfill u&=&0 &\text {on }\partial \Omega \setminus \{ 0 \} , \end{array} \right . \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than s greater-than 2"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>s</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0>s>2</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="2 Superscript star Baseline left-parenthesis s right-parenthesis colon-equal StartFraction 2 left-parenthesis n minus s right-parenthesis Over n minus 2 EndFraction"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>⋆</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>≔</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">{2^\star (s)}≔\frac {2(n-s)}{n-2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".