Peak solutions for logarithmic scalar field systems
Bibliographic record
Abstract
We are concerned with a class of important Schrdinger equations in mathematical physics with logarithmic nonlinearities: - 2 u +V (y)u = u log|u|, u > 0, in H 1 (R N ), where N 3, is a small positive parameter, and V (y) denotes the potential function.The main difficulties to apply Lyapunov-Schmidt reduction to logarithmic scalar equations are caused by the non-smooth property and sublinear growth of the logarithmic non-linearity.Our method is fundamentally based on a new type of innerouter decomposition, setting it apart from conventional gluing techniques that usually require distinct constructions for the inner and outer problems.Rather than this traditional separation, we incorporate the minimization operator for the outer problem directly with the operator related to the fixed-point theorem, enhancing the reduction framework to be applicable.We prove the existence of positive multipeak solutions under certain assumptions on V (y).Finally, we also use the local Pohozaev identities to obtain the non-degenerate of positive multipeak solutions.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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 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".