Liouville Type Theorems for Stable Solutions of Certain Elliptic Systems
Bibliographic record
Abstract
Abstract We establish Liouville type theorems for elliptic systems with various classes of nonlinearities on ℝ N . We show, among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the derivatives of the corresponding non-linearities is positive. We give some immediate applications to various standard systems, such as the Gelfand, and certain Hamiltonian systems. The case where the infimum is zero is more interesting and quite challenging. We show that any C 2 (ℝ N ) positive entire semi-stable solution of the following Lane-Emden system, is necessarily constant, whenever the dimension N < 8 + 3α + , provided p = 1, q ≥ 2 and f (x) = (1 + |x| 2 ) . The same also holds for p = q ≥ 2 provided We also consider the case of bounded domains Ω ⊂ ℝ N , where we extend results of Brown et al. [1] and Tertikas [18] about stable solutions of equations to systems. At the end, we prove a Pohozaev type theorem for certain weighted elliptic systems.
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".