Multi-bump solutions of -Δ<i>n</i> = <i>K</i>(<i>x</i>)<i>u</i> <sup>(<i>n</i>+2)/(<i>n</i>-2)</sup> on lattices in ℝ<sup> <i>n</i> </sup>
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Bibliographic record
Abstract
Abstract We consider the following semilinear elliptic equation with critical exponent: Δ u <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>=</m:mo> </m:math> = K ( x ) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>u</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> u^{(n+2)/(n-2)} , u <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>></m:mo> </m:math> > 0 in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> \mathbb{R}^{n} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> {n\geq 3} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {K>0} is periodic in ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> x_{1} ,…, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> x_{k} ) with 1 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>≤</m:mo> </m:math> \leq k <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo><</m:mo> </m:math> < ( n -2)/2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>k</m:mi> </m:msup> </m:math> {\mathbb{R}^{k}} , including infinite lattices. We also show that for k <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>≥</m:mo> </m:math> \geq ( n -2)/2, no such solutions exist.
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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 | 0.004 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
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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 it