Normalized solutions for a nonlinear Schrödinger equation via a fixed point theorem
Bibliographic record
Abstract
In this paper, we aim to investigate the existence and boundedness of the normalized solutions of the following nonlinear Schrdinger equation by using a non-variational fixed point theorem:where the potential function V (x) : R is measurable and can change sign, h 0 is the perturbation term, the nonlinearity f is measurable and satisfies critical or subcritical growth for N 3, exponential critical growth for N = 2.We first consider as the whole space R N and establish the existence of normalized solutions to the problem for N = 2 and N 3, respectively.While is considered as a bounded domain, we establish the existence and L -boundedness of the normalized solution of the problem for N 3, where the nonlinearity f satisfies the subcritical growth condition.In the bounded domain , we can still obtain the existence of normalized solutions to the above problem with the critical growth condition instead of the L -boundedness of normalized solutions.As far as we know, our results are more general and novel on this topic, and can also provide a new approach for the study of nonlinear Schrdinger equation with prescribed L 2 -norm.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 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".