SEMI-CLASSICAL STATES FOR QUASILINEAR SCHRÖDINGER EQUATIONS ARISING IN PLASMA PHYSICS
Bibliographic record
Abstract
In this paper, the existence and qualitative properties of positive ground state solutions for the following class of Schrödinger equations -ε2Δu + V(x)u - ε2[Δ(u2)]u = f(u) in the whole two-dimensional space are established. We develop a variational method based on a penalization technique and Trudinger–Moser inequality, in a nonstandard Orlicz space context, to build up a one parameter family of classical ground state solutions which concentrates, as the parameter approaches zero, around some point at which the solutions will be localized. The main feature of this paper is that the nonlinearity f is allowed to enjoy the critical exponential growth and also the presence of the second order nonhomogeneous term -ε2[Δ(u2)]u which prevents us from working in a classical Sobolev space. Our analysis shows the importance of the role played by the parameter ε for which is motivated by mathematical models in physics. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".