Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone
Pith reviewed 2026-05-25 18:27 UTC · model grok-4.3
The pith
Nonradial positive and nodal solutions exist for the critical Neumann problem in cones when the angular domain is symmetric or has large enough volume.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If ω is symmetric with respect to the north pole of S^{N-1}, the problem admits a nonradial sign-changing solution. If the volume of Σ_ω ∩ B_1(0) is large enough, though possibly smaller than half the volume of B_1(0), the problem admits a positive nonradial solution. Both conclusions require the local convexity condition at the boundary of the cone.
What carries the argument
Variational arguments on the energy functional of the critical Neumann problem, applied separately under symmetry constraints on ω and under volume comparisons on the truncated cone.
If this is right
- Nonradial sign-changing solutions are guaranteed whenever the angular domain ω is symmetric about a pole.
- Positive nonradial solutions exist for cones whose truncated volume exceeds a positive threshold that need not reach half the ball volume.
- The local convexity condition is sufficient to make the symmetry-breaking arguments work in both cases.
Where Pith is reading between the lines
- The volume threshold might be improved or shown sharp by testing on explicit rotationally symmetric cones such as half-spaces.
- Similar existence statements could be sought for the same equation under Dirichlet instead of Neumann boundary conditions.
- The results suggest that partial symmetry in the domain is often enough to produce extra solutions at the critical exponent.
Load-bearing premise
The local convexity condition at the boundary of the cone must hold, otherwise the variational arguments used to obtain the nonradial solutions may fail.
What would settle it
An explicit example of a symmetric ω for which the problem has no nonradial sign-changing solution, or a cone with truncated volume above the threshold that has only radial positive solutions, would disprove the claims.
read the original abstract
We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded cone $\Sigma_\omega:=\{tx:x\in\omega\text{ and }t>0\}$, where $\omega$ is an open connected subset of the unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$ with smooth boundary, $N\geq 3$ and $2^*:=\frac{2N}{N-2}$. We assume that some local convexity condition at the boundary of the cone is satisfied. If $\omega$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone $\Sigma_\omega\cap B_1(0)$ is large enough (but possibly smaller than half the volume of the unit ball $B_1(0)$ in $\mathbb{R}^N$), we establish the existence of a positive nonradial solution.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the critical Neumann problem -Δu = |u|^{2*-2}u in the unbounded cone Σ_ω with Neumann boundary condition, under a local convexity assumption on the cone. It proves existence of a nonradial sign-changing solution when ω is symmetric with respect to the north pole of S^{N-1}, and existence of a positive nonradial solution when the volume of Σ_ω ∩ B_1(0) is sufficiently large (possibly less than half the ball volume). The proofs rely on variational methods, specifically mountain-pass and genus arguments on the Nehari manifold, with compactness restored via a boundary trace inequality from local convexity and test-function constructions exploiting symmetry or volume.
Significance. If the results hold, they provide new existence theorems for nonradial solutions to critical problems in conical domains, extending standard variational techniques (Nehari manifold, concentration-compactness) to this geometry. The explicit use of symmetry to work in an invariant subspace and volume to produce subcritical-energy test functions are constructive strengths that could inform related problems in symmetric or large-volume domains.
minor comments (3)
- The abstract and introduction should explicitly state the precise form of the local convexity condition (e.g., the inequality involving the second fundamental form or mean curvature) rather than referring to it only as 'satisfied,' to allow readers to verify applicability without consulting prior works.
- In the concentration-compactness argument (likely §4 or §5), clarify how the cone geometry modifies the standard Lions lemma; a brief remark on the adaptation of the profile decomposition to the unbounded conical setting would improve readability.
- Notation for the unitary bounded cone Σ_ω ∩ B_1(0) is used without a dedicated definition or figure; adding a short paragraph or diagram in §2 would help distinguish it from the full cone Σ_ω.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript, including the summary of the main results on nonradial positive and nodal solutions to the critical Neumann problem in cones, the significance of the variational approach, and the recommendation for minor revision. No major comments were provided in the report.
Circularity Check
No significant circularity in derivation chain
full rationale
The paper is a standard existence proof for a critical Neumann problem in a cone. It deploys mountain-pass and genus arguments on the Nehari manifold, recovers the Palais-Smale condition via concentration-compactness adapted to the cone, and uses the symmetry assumption on ω to work in an invariant subspace and the volume condition to produce an explicit test function whose energy lies strictly below the Sobolev threshold. The local convexity hypothesis is an explicit assumption that yields a boundary trace inequality restoring compactness. None of these steps reduces by definition, by fitted-parameter renaming, or by a self-citation chain to the target existence statement; all constructions are carried out directly from the variational functional and the given geometric hypotheses. The result is therefore self-contained against external benchmarks and receives score 0.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard Sobolev embeddings and critical point theory (mountain-pass or minimax) apply to the energy functional on the cone.
Reference graph
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discussion (0)
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