Mixed local and nonlocal supercritical Dirichlet problems
Bibliographic record
Abstract
In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem $ \begin{equation} Lu = |u|^{p-2}u+\mu |u|^{q-2}u\quad\text{in}\; \; \Omega, \quad\quad u = 0\quad\text{in}\; \; \mathbb{R}^N\setminus\Omega, ~~~(1) \end{equation} $ where $ L: = -\Delta +(-\Delta)^s $ for $ s\in(0, 1) $ and $ \Omega\subset\mathbb{R}^N $ is a bounded domain. Precisely, we show that problem (1) for $ 1<q<2<p $ has a positive solution as well as a sequence of sign-changing solutions with a negative energy for small values of $ \mu $. Here $ u $ can be either a scalar function, or a vector valued function so that (1) turns into a system with supercritical nonlinearity. Moreover, whenever the domain is symmetric, we also prove the existence of symmetric solutions enjoying the same symmetry properties. We shall also prove an existence result for the supercritical Hamiltonian system$ Lu = |v|^{p-2}v, \qquad Lv = |u|^{d-2}u+\mu |u|^{q-2}u $with the Dirichlet boundary condition on $ \Omega $ where $ 1<q<2
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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 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.002 |
| 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".