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Critical quasi-linear Schr\"{o}dinger system with $p$-Laplacian

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Pith's one-line read Positive solutions to the critical quasi-linear Schrödinger system with p-Laplacian are radially symmetric, unique up to translation, and fully classified.

desk verdict Extends uniqueness and classification of positive solutions from the Laplacian to the p-Laplacian case, but the explicit criticality relation for α and β is missing from the abstract. read the letter →

arxiv 2606.10630 v1 pith:2KC3QPFP submitted 2026-06-09 math.AP

Neng cheng , Wei Dai , Zhao Liu This is my paper · ORCID
classification math.AP
keywords p-Laplacianquasi-linearSchrödingersystemcriticalexponentradialsymmetryuniquenesspositivesolutionsD^{1p}spacemovingplanes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies positive solutions to a system of quasilinear equations driven by the p-Laplacian at the precise critical exponent. It first establishes regularity of solutions together with sharp decay estimates at infinity. It then proves that every positive solution pair must be radially symmetric and strictly decreasing away from a single point in space. Finally it shows that all such solutions are equivalent under translation, giving a complete classification that extends earlier uniqueness theorems known only for the linear Laplacian case p equals 2.

What carries the argument

Regularity theory combined with sharp asymptotic estimates followed by the method of moving planes to establish symmetry and uniqueness for the critical system.

What would settle it

Exhibiting either a positive solution that fails to be radially symmetric about any point or two positive solutions not related by a spatial translation would contradict the classification.

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Extended reading notes

Core claim

We obtain the uniqueness and complete classification of positive solutions to the D^{1,p}(R^N)-critical quasi-linear Schrödinger system with p-Laplacian for 1 < p < N. All positive solutions are radially symmetric and strictly decreasing about some point, extending the corresponding uniqueness results known for the case p = 2.

Load-bearing premise

The exponents alpha and beta must satisfy the exact scaling relation that makes the nonlinearity critical with respect to the D^{1,p} Sobolev norm.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 0 minor

Summary. The paper studies positive solutions (u,v) in D^{1,p}(R^N) to the quasilinear system -Δ_p u = u^α v^β, -Δ_p v = u^β v^α in R^N, under the assumptions 1 < p < N, N ≥ 2, 0 ≤ α ≤ β. It claims to prove regularity, sharp asymptotic estimates at infinity, radial symmetry and strict monotonicity about some point via moving planes or similar, and finally uniqueness together with a complete classification of all such positive solutions, extending the p=2 results of [LM,QS].

Significance. If the derivations hold, the work would deliver a full classification of positive solutions for the D^{1,p}-critical quasilinear Schrödinger system, extending the Laplacian case to the p-Laplacian setting. This is a substantive contribution to the literature on critical elliptic systems, provided the scaling-critical relation is correctly identified and the Pohozaev identity closes.

major comments (2)
  1. [Abstract] Abstract (and presumably the setup in §1 or §2): the system is asserted to be D^{1,p}-critical, yet the explicit algebraic relation between α, β, p and N that enforces scale invariance under the D^{1,p} norm (typically of the form α + β = (N(p-1) + p)/(N-p) or the system-adjusted critical exponent) is never displayed. Without this relation the Pohozaev-type identity and the asymptotic matching used for classification cannot be verified to close, rendering the uniqueness claim unverifiable from the given information.
  2. [Abstract] The range 0 ≤ α ≤ β is stated without confirmation that it is compatible with the critical scaling; if the relation in the previous comment is not satisfied inside this range, the moving-plane argument and the claimed radial symmetry may fail to apply.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to explicitly display the criticality condition. We agree that this should be stated clearly and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and presumably the setup in §1 or §2): the system is asserted to be D^{1,p}-critical, yet the explicit algebraic relation between α, β, p and N that enforces scale invariance under the D^{1,p} norm (typically of the form α + β = (N(p-1) + p)/(N-p) or the system-adjusted critical exponent) is never displayed. Without this relation the Pohozaev-type identity and the asymptotic matching used for classification cannot be verified to close, rendering the uniqueness claim unverifiable from the given information.

    Authors: We thank the referee for this observation. The D^{1,p}-critical condition for the system is α + β = \frac{N(p-1) + p}{N - p}. This relation ensures the right-hand sides are homogeneous of the correct degree with respect to the D^{1,p} scaling, so that the Pohozaev identity closes and the asymptotic decay rates are consistent with the classification. Although the proofs are carried out under this scaling, we acknowledge the relation was not written explicitly in the abstract or early sections. We will insert the formula prominently in the revised abstract and §1. revision: yes

  2. Referee: [Abstract] The range 0 ≤ α ≤ β is stated without confirmation that it is compatible with the critical scaling; if the relation in the previous comment is not satisfied inside this range, the moving-plane argument and the claimed radial symmetry may fail to apply.

    Authors: Once the critical relation α + β = \frac{N(p-1) + p}{N - p} is fixed, the ordering 0 ≤ α ≤ β is without loss of generality by symmetry of the system in (u,v). This range lies inside the admissible set for the critical exponent and preserves the cooperative structure needed for the moving-plane method: the map (s,t) ↦ s^α t^β remains positive and increasing in each variable separately. The proofs in §§4–5 verify the required monotonicity conditions directly under this ordering. We will add a short remark after the statement of the critical relation confirming compatibility. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: classification derived from standard regularity, symmetry, and Pohozaev analysis rather than self-definition or fitted inputs

full rationale

The paper establishes regularity, asymptotic decay, radial symmetry, and uniqueness for positive solutions of the stated system by extending known p=2 techniques to the p-Laplacian case via functional-analytic methods. The criticality assertion is used to set up the problem but does not reduce the classification result to a tautology or to a parameter fitted from the target solutions themselves. No self-citation chain is load-bearing for the central uniqueness claim, and the derivation does not rename a known empirical pattern or smuggle an ansatz. The result is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claims rest on standard functional analysis for the p-Laplacian and Sobolev embeddings together with the assumption that the system is exactly critical; no free parameters or new entities are introduced in the abstract.

assumptions (2)
  • standard math Standard properties of the p-Laplacian and the space D^{1,p}(R^N) (Sobolev embeddings, regularity theory)
    Invoked throughout to obtain regularity and asymptotic behavior.
  • domain assumption The nonlinearity is D^{1,p}-critical
    The title and abstract label the system critical, which fixes the relation between α, β and p.

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Pith. "Pith review of Critical quasi-linear Schr\"{o}dinger system with $p$-Laplacian." pith.science (2026). https://pith.science/paper/2KC3QPFP

@misc{pith2026260610630,
  author       = {Pith},
  title        = {Pith review of: Critical quasi-linear Schr\"odinger system with $p$-Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2KC3QPFP}},
  note         = {Machine review of arXiv:2606.10630}
}
abstract

In this paper, we mainly consider positive solution to the $D^{1,p}(\R^{N})$-critical quasi-linear Schr\"{o}dinger system with $p$-Laplacian: \begin{equation*}\begin{cases} -\Delta_p u = u^{\alpha}v^{\beta} \, \ \ \ \ \ \text{in}\,\ \ \R^N, \\ -\Delta_p v = u^{\beta}v^{\alpha} \,\ \ \ \ \ \text{in}\,\ \ \R^N, \end{cases}\end{equation*} where $1<p<N$, $N\geq2$, $0\leq \alpha \leq \beta,$ and $u,v\in D^{1,p}(\R^N)$. We establish regularity and the sharp estimates on asymptotic behaviors for any positive solution $(u,v)$. Then, we prove that all positive solutions are radially symmetric and strictly decreasing about some point. Furthermore, we obtain the uniqueness and complete classification of positive solutions. Our results extend the uniqueness results in \cite{LM,QS} for $p=2$ to general cases $1<p<N$.

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