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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 source (direct Gemma or distilled Codex), 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".