The number of positive solutions for n$n$‐coupled elliptic systems
Bibliographic record
Abstract
Abstract We study the number of positive solutions to the ‐coupled elliptic system where , , , , , and for . We prove new multiplicity and uniqueness results for positive solutions of the system, whether the system has a variational structure or not. In some cases, we provide a rather complete characterization on the exact number of positive solutions. The results we obtain reveal that the positive solution set of this system has very different structures in the three cases , , and . Moreover, when , very different structures of the positive solution set can also be seen in the case where close to 2 and the case where close to . Similar results are given for elliptic systems with subcritical Sobolev exponents. These results substantially generalize and improve existing results in the literature. To show the effect of the uniqueness result, we apply it to prove existence of a positive solution to a 2‐coupled nonlinear Schrödinger system with critical exponent and potentials.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| 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 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".