A New Variational Principle and Supercritical Semilinear Elliptic Problems
Bibliographic record
Abstract
Our objective in this thesis is to study semilinear elliptic partial differential equations when the nonlinearity may have supercritical Sobolev growth. We shall apply a new variational principle recently introduced in [25, 26] to prove the existence of solutions for such problems. We would like to emphasize that functionals that we are using in the new variational principle are different from the standard Euler-Lagrange functionals that are mostly used in the literature. The results in this thesis are twofold. Namely, we first prove the existence of a solution for the semilinear elliptic equations in the presence of a subsolution and a supersolution. Secondly, we consider the general case, and discuss the existence and smoothness of its solutions in a supercritcal case. We remark that the latter results are recently published in [9].
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| 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) | 0.004 | 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".