A Neumann problem with supercritical and exponential growth in dimension N ≥ 3
Bibliographic record
Abstract
Abstract We will consider the following problem: -Δu + u = f(u), x ∈ Ω, u > 0, x ∈ Ω, ∂u/∂ν = 0, x ∈ ∂Ω, where Ω is a bounded annulus in ℝ^N (N ≥ 3), and the function f has either exponential growth given by the Trudinger-Moser inequality f(u) = u(e^{u^2}-1), or it is of the power form f(u) = u|u|^{p-2}, where p is supercritical. It is known that this problem always has a positive radial solution. Our main goal in this paper is to prove the existence of a positive non-radial solution for the case f(u) = u(e^{u^2}-1), and to establish the multiplicity of non-radial positive solutions for the case f(u) = u|u|^{p-2} when the annulus is thin. We shall first state our results for a general annulus when the right hand side of equation is of the form a(x) f(u) where the function a(x) belongs to a class of sufficiently smooth non-negative functions which enjoys certainsymmetry and monotonicity properties. This class includes the case where the function a is radial.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| 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".