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 Schrödinger equation by using a non-variational fixed point theorem:where the potential function V (x) : Ω → 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 Ω as the whole space R N and establish the existence of normalized solutions to the problem for N = 2 and N ≥ 3, respectively.While Ω 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 Ω, 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 Schrödinger equation with prescribed L 2 -norm.
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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 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".