On a Fourth Order Elliptic Problem with a Singular Nonlinearity
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Bibliographic record
Abstract
Abstract We study the following semilinear biharmonic equation: , where 0 ≤ f ≤ 1 and B R ⊂ ℝ N , N ≥ 1, is the ball centered in the origin of radius R. We prove, under Dirichlet boundary conditions u = Әu/Әƞ = 0 on ӘB R , the existence of λ* = λ*(R, f) > 0 such that for λ ∈ (0, λ*) there exists a minimal (classical) solution u̲ λ < which satisfies 0 < u̲ λ < 1. In the extremal case λ = λ*, we prove the existence of a weak solution which has finite energy and which is the unique solution even in a very weak sense. For λ > λ* there are no solutions of any kind. Estimates on λ*, stability properties of solutions and nonexistence results in the whole space are also established.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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 it