Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation
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Bibliographic record
Abstract
We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations \Delta u - u + f(u) =0 in \mathbb R^N , u \in H^1 (\mathbb R^N) , where N\geq 2 . Under natural conditions on the nonlinearity f , we prove the existence of infinitely many nonradial solutions in any dimension N \geq 2 . Our result complements earlier works of Bartsch and Willem ( N=4 or N \geq 6 ) and Lorca-Ubilla ( N=5 ) where solutions invariant under the action of O(2) \times O(N-2) are constructed. In contrast, the solutions we construct are invariant under the action of D_k \times O(N-2) where D_k \subset O(2) denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with k sides invariant, for some integer k\geq 7 , but they are not invariant under the action of O(2) \times O(N-2) .
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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 | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it