Arbitrary Number of Positive Solutions For an Elliptic Problem with Critical Nonlinearity
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Bibliographic record
Abstract
We show that the critical nonlinear elliptic Neumann problem \Delta u -\mu u + u^{7/3} = 0 \text{ in } \Omega,\ u >0 \ \text{ in } \Omega \text{ and } \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial \Omega where \Omega is a bounded and smooth domain in \R^5 , has arbitrarily many solutions, provided that \mu>0 is small enough. More precisely, for any positive integer K , there exists \mu_K >0 such that for 0 <\mu < \mu_K , the above problem has a nontrivial solution which blows up at K interior points in \Omega , as \mu \to 0 . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.
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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 | 0.001 | 0.000 |
| 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 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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