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Uniqueness of positive solutions for finsler p-Laplacian equations with polynomial non-linearity

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Pith's one-line read This paper proves that a Finsler p-Laplacian Dirichlet problem has at most one positive solution when 3/2 < p < 2 and the exponent q sits just above p−1.

desk verdict The paper makes a real advance in the singular Finsler p-Laplacian regime, but Theorem 1.2's final contradiction is a non sequitur as written. read the letter →

arxiv 2411.15227 v2 pith:JAJWB5XV submitted 2024-11-21 math.AP

Rongxun He , Wei Ke This is my paper · ORCID
classification math.AP MSC 35J9235A0235B4535J60
keywords Finslerp-LaplaciananisotropicellipticequationsuniquenessofpositivesolutionslinearizedmethodweightedSobolevembeddingHardyinequalitysingularcaseLane-Emdenproblem
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

This paper proves that the anisotropic Finsler p-Laplacian Dirichlet problem $-\Delta^F_p u = u^q$ with zero boundary data has at most one positive solution on a bounded $C^2$ domain when $3/2 < p < 2$ and $q$ is close to $p-1$ from above. The near-critical regime needs no restriction on the anisotropy: uniqueness is forced by the structure of the equation. A companion perturbation statement says that for each fixed admissible $q$, if the Euclidean p-Laplace problem has a unique non-degenerate positive solution, then every Finsler norm sufficiently close to the Euclidean norm in $C^{3,\alpha}$ on the unit sphere inherits uniqueness. This moves a previously known ground-state uniqueness result to all positive solutions and into the singular exponent range $p<2$.

What carries the argument

The engine is the linearized method: normalize two hypothetical positive solutions by their $L^\infty$ maxima, form their sign-changing difference $w_k$, and show that $w_k$ converges to a first eigenfunction of the weighted linear eigenvalue problem $\lambda(\Omega; U) = \inf \int_\Omega \langle D^2H(\nabla U)\nabla\varphi, \nabla\varphi\rangle \, dx \big/ \int_\Omega U^{p-2}\varphi^2\, dx$, whose only minimizers are $\pm U$ and whose value equals $p(p-1)\lambda_F(\Omega)$. Two imported regularity inputs carry the argument: a uniform $C^{1,\gamma}$ bound together with a boundary-layer gradient lower bound $|\nabla u| \ge \mu_0$ for normalized solutions, and the integrability of $|\nabla u|^{-r}$ for every $r < p-1$. These convert the classical Hardy inequality into a uniform weighted compact embedding, which is what lets the sequence of normalized differences pass to the limit.

What would settle it

Compute or numerically test the boundary gradient of normalized solutions for a family of $C^{3,\alpha}$-near-Euclidean Finsler norms with $p<2$ and $q$ approaching $p-1$: if for some admissible domain and exponent the infimum of $|\nabla u|$ on the boundary layer tends to zero, the uniform lower bound behind the compactness step is false. Alternatively, exhibit one Finsler norm $F$ and one exponent $q$ in the stated interval with two distinct positive solutions, for example on a $C^2$ domain where the Euclidean problem already admits multiple positive solutions.

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Extended reading notes

Core claim

The central claim is that, under the stated hypotheses, the normalized difference of any two supposed positive solutions leads to a contradiction: it must change sign by a Picone-type comparison, but the linearized equation forces its limit to be a first eigenfunction of a weighted linear eigenvalue problem, and first eigenfunctions have one sign. The proof blows up solutions about their maximum points, uses a boundary gradient lower bound together with Hardy's inequality to obtain a uniform weighted compact embedding relative to $|\nabla \tilde u|$, and then passes to the limit as $q_k \to p-1$. The limit function $w$ solves $-\operatorname{div}(D^2H(\nabla \tilde u)\nabla w) = p(p-1)\lambda_F(\Omega)\, \tilde u^{p-2} w$, and the only attainable minimizers of this weighted problem are $\pm \tilde u$, so the sign contradiction is unavoidable. The restriction $p > 3/2$ enters because the proof needs the inverse-gradient integrability estimate $|\nabla u|^{-r} \in L^1$ for $r = 2-p$, which requires $r < p-1$.

Load-bearing premise

The proof leans on an imported regularity estimate: any solution normalized by its maximum has a uniformly controlled first derivative and, near the boundary, a gradient size bounded away from zero, with constants independent of the exponent; if that boundary lower bound ever fails, the Hardy-type compactness step that drives the contradiction collapses.

Editorial extensions

If this is right

  • For every bounded $C^2$ domain $\Omega$ and $3/2 < p < 2$, there is $q_0 = q_0(N,\Omega,p) > p-1$ such that the anisotropic Lane-Emden problem has exactly one positive solution whenever $p-1 < q < q_0$.
  • The earlier ground-state-only uniqueness result for the isotropic p-Laplacian is upgraded to uniqueness of all positive solutions, and the exponent range now includes the singular p-Laplacian case $p<2$.
  • Near the Euclidean norm in $C^{3,\alpha}(S^{N-1})$, uniqueness persists for each fixed $q \in (p-1, Q_{0,p})$ whenever the Euclidean problem has a unique non-degenerate positive solution.
  • Along any sequence $q_k \to p-1$, normalized solutions converge in $C^{1,\beta}$ to the first eigenfunction of $-\Delta^F_p$, and the normalization constants $M_k^{q_k+1-p}$ converge to the first eigenvalue $\lambda_F(\Omega)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The open interval $1 < p \le 3/2$ is a natural target for a counterexample search: the proof uses $p > 3/2$ only to make $r = 2-p$ admissible in the inverse-gradient integrability estimate, so any failure of uniqueness on that interval would most plausibly stem from exactly that input.
  • Theorem 1.1 suggests that uniqueness is structurally stable under small anisotropic perturbations of the Euclidean norm; a quantitative question follows naturally, namely how large the admissible $C^{3,\alpha}$ distance $\delta_1$ can be for a fixed $q$, and whether uniqueness persists away from the near-critical regime.
  • The Hardy-type weighted compact embedding built in the proof depends only on the uniform boundary gradient lower bound, not on the specific polynomial form $u^q$; it could therefore be reused for nonlinearities with boundary singularities or for systems driven by the same anisotropic operator.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies uniqueness of positive solutions to the Finsler p-Laplacian Dirichlet problem -Δ_F^p u = u^q in Ω, u=0 on ∂Ω, for 3/2 < p < 2 and q > p-1. Theorem 1.1 claims uniqueness for q in (p-1, Q_{0,p}) when the Finsler norm F is sufficiently close to the Euclidean norm, assuming the Euclidean problem has a unique non-degenerate positive solution. Theorem 1.2 claims, for every bounded C^2 domain and every admissible F, the existence of q_0 > p-1 such that uniqueness holds for all q in (p-1, q_0). The proof linearizes around two hypothetical distinct solutions, normalizes their difference in a weighted L^2 norm, passes to a limit in a weighted Sobolev space using a Hardy-type embedding, and aims for a contradiction by showing the limit would be a sign-definite first eigenfunction of a weighted eigenvalue problem, while the approximating sign-changing differences have both sign sets of uniformly positive measure.

Significance. If the main results are correct, the paper would substantially extend the uniqueness theory of Brasco and Lindgren from the ground-state setting and the range p>2 to all positive solutions and the singular range 3/2 < p < 2, in the anisotropic Finsler framework. The paper contains a number of well-structured components: a blow-up argument for uniform L∞ bounds, a Hardy-type uniform weighted embedding (Theorem 2.2), a detailed treatment of the weighted eigenvalue problem (Section 3), and a clean use of non-degeneracy in Theorem 1.1. The proofs of Propositions 3.1 and 3.2 are essentially complete and the overall strategy is coherent. However, the final contradiction in Theorem 1.2 is not logically justified as written, and this is load-bearing for the paper's central extension.

major comments (3)
  1. [§4, Step 3 (after Eq. (4.16))] The final contradiction is a non sequitur. The authors prove only the measure lower bound |Ω_k^±| ≥ 1/C' for the supports of the positive and negative parts of w_k, and then assert that this contradicts the strong L^2 convergence of w_k to a sign-definite nontrivial limit w. But a measure lower bound is compatible with collapse of the L^2 mass: for instance, in any bounded domain, w_k = 1 - k^{-1} χ_E with |E| ≥ c satisfies |{w_k < 0}| ≥ c while w_k → 1 strongly in L^2. No uniform lower bound on ||w_k^±||_{L^2} or on the corresponding weighted L^2 norm is derived in Step 3; equation (4.16) only relates the weighted norm of a sign part to its energy, and both can tend to zero simultaneously. Since this is the only step that excludes the existence of two distinct positive solutions, the proof of Theorem 1.2 is incomplete as written.
  2. [Theorem 2.2 and Corollary 2.1] The uniform Hardy-type inequality is proved for N > 2, since the exponent σ0 = 2 + 2(2+s0)/(N-2) is undefined for N = 2 and the Sobolev exponent 2^* = 2N/(N-2) is infinite in that case. The main theorems allow N = 2, and Step 3 uses the measure bound derived from Theorem 2.2 for some 2 < σ < σ0. The gap is probably fixable by taking σ0 arbitrarily large and using the two-dimensional Sobolev embedding for finite exponents, but the N = 2 case is not covered by the proof as written.
  3. [Theorem 2.1] The uniform C^{1,γ} estimates and, in particular, the boundary-layer gradient lower bound |∇u_{F,q}| ≥ μ0 in Ω_τ are imported from [4, Theorem 2.5] with [18]. The paper states that these results apply for all p > 1, but the cited [4] concerns the case p > 2, while the present application is 3/2 < p < 2. Since Theorem 2.2, Corollary 2.1, and the compactness passage in Step 2 all depend on this uniformity, the authors should either provide a proof for the anisotropic p < 2 setting or give a precise reference that covers it. This is a missing justification for a load-bearing input, not merely a citation issue.
minor comments (5)
  1. [Introduction and Theorem 1.1] The exponent Q_{0,p} is not defined in the paper; it is only said to be defined in [2, Section 4]. The range q ∈ (p-1, Q_{0,p}) should be stated explicitly or the relevant condition should be reproduced.
  2. [Throughout] There are several typos and formatting issues: 'finslerp-Laplacian' in section headings, 'largerly' for 'largely' in Section 1.3, and inconsistent uses of 'Hölder' vs. 'Holder' in the text.
  3. [Theorem 2.2 proof] The proof explicitly treats only the case s < 0; the case s ≥ 0 is dismissed with 'We only deal with the case s < 0'. A short sentence explaining why the other case is easier would improve readability.
  4. [§4, Step 2] The convergence of A_k to A∞ in Z_δ^c requires a dominated-convergence argument in the t-integration when μ' = 0, because the integrand D^2H(t∇ũ_k + (1-t)∇ṽ_k) is singular near t = 0. The paper does not spell this out; the statement 'A_k → A∞ uniformly in Z_δ^c' is not immediate.
  5. [Corollary 2.1 and Eq. (4.12)] The derivation of (4.12) for the double integral weight ∫_0^1 (tũ_k+(1-t)ṽ_k)^{q_k-1} dt is only sketched. Since this weight is not literally of the form |u_k|^{q_k-1}, a few lines explaining the reduction to Theorem 2.2 would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all load-bearing inputs are external, the normalization is not a fitted parameter, and the Euclidean uniqueness assumption in Theorem 1.1 is an explicit hypothesis.

full rationale

The paper's central claim, uniqueness of all positive solutions to the Finsler p-Laplacian problem for 3/2<p<2 and q close to p-1, is proved by a linearization/compactness argument rather than by importing the desired conclusion. The load-bearing regularity inputs, Theorem 2.1 and Theorem 2.3, are quoted from external works [4, 8, 9, 18] under stated hypotheses that do not include the target uniqueness; they are not results of the present authors and do not depend on the paper's own fitted values. The weighted normalization S_k in (4.6) is a natural scaling of the difference of two solutions, not a parameter fitted to force the contradiction; the limit equation for w is obtained by passing to the limit in the linearized equation, not assumed. The p>3/2 condition arises from applying Theorem 2.3 with r=2-p, an external integrability estimate, not from any circular construction. In Theorem 1.1, the uniqueness and nondegeneracy of the Euclidean problem are explicit hypotheses of a perturbation statement, so using them is an honest assumption rather than a hidden restatement of the conclusion. The only self-citation, [16] by author Ke, appears as background in the introduction and is never used as a load-bearing premise; the uniqueness theorem is not imported from the authors' own prior work. The final contradiction in Step 3, whether or not it is logically airtight, is a mathematical correctness concern about measure lower bounds versus L^2 convergence, not an instance of circularity: it does not reduce the conclusion to its inputs by definition or by construction. Accordingly, the derivation chain is self-contained relative to external benchmarks and no circularity is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

This paper proves two theorems, so the ledger contains no fitted free parameters and no invented entities. What the central claim rests on is a stack of imported results: Theorem 2.1 (uniform C^{1,gamma} estimates and boundary gradient lower bound) taken from [4, Theorem 2.5] with [18]; Theorem 2.3 (integrability of |grad u|^{-r} for r < p-1) taken from [8, 9]; the Liouville theorems of [2, Theorems 3.1 and 4.4] used in Lemma 5.1; and the usual existence, comparison, and Picone toolkit for the Finsler p-Laplacian. The only genuinely new technical node is Section 2.2's uniform weighted compact embedding, built on the Hardy inequality and Theorem 2.1(2). Counting these imports is the honest measure of the paper's marginal contribution: a careful adaptation of a known scheme to the singular range, conditional on the cited regularity theory being correct in the stated uniformity.

assumptions (5)
  • domain assumption Uniform C^{1,gamma} a priori estimates and boundary gradient lower bound |grad u_{F,q}| >= mu_0 on the boundary layer Omega_tau, uniformly over F near the Euclidean norm or q near p-1.
    Invoked as Theorem 2.1 and proven by referencing [4, Theorem 2.5] and [18, Theorem 1]. It is the regularity input to the Hardy-type inequality (Theorem 2.2) and to the a priori bounds in Lemma 2.3.
  • domain assumption Weighted integrability of the inverse gradient: sup_x integral_Omega |grad u(y)|^{-r} |x-y|^{-gamma} dy <= S_tilde for all r < p-1 (Theorem 2.3).
    Attributed to [8] (anisotropic case) and [9] (p-Laplacian). Taking r = 2-p and using r < p-1 gives 3/2 < p, the stated range of Theorem 1.2.
  • standard math Liouville theorem for -Delta_p^F u = u^{p-1} in R^N and in the half-space R^N_+ with zero Dirichlet data.
    Proved sketchily in Lemma 2.2 via the eigenvalue test with the first Finsler p-Laplacian eigenvalue on balls, lambda_R going to 0; the test-function argument uses psi_R = phi_R^p u^{1-p} and requires truncation and regularity details that are not written out.
  • domain assumption Euclidean Liouville theorems for -Delta_p u = u^q in R^N and R^N_+ when q < Q_{0,p} (Bidaut-Veron and Pohozaev).
    Imported from [2, Theorems 3.1 and 4.4]; Q_{0,p} is defined in [2, Section 4]. Used in Lemma 5.1 for the uniform L^infinity bound in the perturbation theorem (Theorem 1.1).
  • domain assumption Existence of at least one positive solution of (1.1) for q in (p-1, p^*-1), standard C^{1,gamma} regularity, and the Picone inequality and comparison principle for the Finsler p-Laplacian.
    Not proved in the paper; taken as background. The Picone inequality is cited from [3, Proposition 2.9] and used in Lemma 2.4 and Proposition 3.2.

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Pith. "Pith review of Uniqueness of positive solutions for finsler p-Laplacian equations with polynomial non-linearity." pith.science (2026). https://pith.science/paper/JAJWB5XV

@misc{pith2026241115227,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of positive solutions for finsler p-Laplacian equations with polynomial non-linearity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAJWB5XV}},
  note         = {Machine review of arXiv:2411.15227}
}
abstract

We consider the uniqueness of the following positive solutions of anisotropic elliptic equation: \begin{equation} \nonumber \left\{ \begin{aligned} -\Delta^F_p u&=u^q \quad \text{in} \quad \Omega, u&=0 \quad \text{on} \quad \partial \Omega, \end{aligned} \right. \end{equation} where $p>\frac{3}{2}$ is a constant. We utilize the linearized method to derive the uniqueness results, which extends the conclusion obtained by L. Brasco and E. Lindgren.

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Works this paper leans on

21 extracted references · 20 canonical work pages

  1. [18]

    G. M. Lieberman. Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal., 12(11):1203–1219, 1988

    1988

  2. [4]

    Brasco and E

    L. Brasco and E. Lindgren. Uniqueness of extremals for some sharp Poincar´ e-Sobolev constants.Trans. Amer. Math. Soc., 376(5):3541–3584, 2023

    work page 2023

  3. [1]

    Adimurthi and S. L. Yadava. An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem.Arch. Rational Mech. Anal., 127(3):219– 229, 1994

    work page 1994

  4. [2]

    M. F. Bidaut-V´ eron and S. Pohozaev. Nonexistence results and estimates for some nonlinear elliptic problems.J. Anal. Math., 84:1–49, 2001

    work page 2001

  5. [3]

    Brasco and G

    L. Brasco and G. Franzina. Convexity properties of dirichlet integrals and picone-type inequalities.Kodai Mathematical Journal, 37(3):769–799, 2014

    work page 2014

  6. [5]

    Brasco, F

    L. Brasco, F. Prinari, and A. C. Zagati. A comparison principle for the Lane-Emden equation and applications to geometric estimates.Nonlinear Anal., 220:Paper No. 112847, 41, 2022

    work page 2022

  7. [6]

    Brezis and M

    H. Brezis and M. Marcus. Hardy’s inequalities revisited.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25(1-2):217–237, 1997

    work page 1997

  8. [7]

    Brezis and L

    H. Brezis and L. Oswald. Remarks on sublinear elliptic equations.Nonlinear Anal., 10(1):55–64, 1986

    work page 1986

Show all 21 references
  1. [8]

    Castorina, G

    D. Castorina, G. Riey, and B. Sciunzi. Hopf Lemma and regularity results for quasilin- ear anisotropic elliptic equations.Calc. Var. Partial Differential Equations, 58(3):Pa- per No. 95, 18, 2019

    2019

  2. [9]

    Damascelli and B

    L. Damascelli and B. Sciunzi. Regularity, monotonicity and symmetry of positive solutions ofm-Laplace equations.J. Differential Equations, 206(2):483–515, 2004. 23

    2004

  3. [10]

    E. N. Dancer. The effect of domain shape on the number of positive solutions of certain nonlinear equations.J. Differential Equations, 74(1):120–156, 1988

    1988

  4. [11]

    De Marchis, M

    F. De Marchis, M. Grossi, I. Ianni, and F. Pacella. Morse index and uniqueness of positive solutions of the Lane-Emden problem in planar domains.J. Math. Pures Appl. (9), 128:339–378, 2019

    2019

  5. [12]

    J. I. D ´ ıaz and J. E. Sa´ a. Existence et unicit´ e de solutions positives pour certaines ´ equations elliptiques quasilin´ eaires.C. R. Acad. Sci. Paris S´ er. I Math., 305(12):521– 524, 1987

    1987

  6. [13]

    DiBenedetto.C 1+α local regularity of weak solutions of degenerate elliptic equa- tions.Nonlinear Anal., 7(8):827–850, 1983

    E. DiBenedetto.C 1+α local regularity of weak solutions of degenerate elliptic equa- tions.Nonlinear Anal., 7(8):827–850, 1983

    1983

  7. [14]

    Gidas, W

    B. Gidas, W. Ni, and L. Nirenberg. Symmetry and related properties via the maxi- mum principle.Comm. Math. Phys., 68(3):209–243, 1979

    1979

  8. [15]

    Han.Nonlinear elliptic equations of the second order, volume 171

    Q. Han.Nonlinear elliptic equations of the second order, volume 171. American Mathematical Soc., 2016

    2016

  9. [16]

    W. Ke. Uniqueness of positive solutions form-Laplacian equations with polynomial non-linearity.Chinese Ann. Math. Ser. B, 46(3):373–406, 2025

    2025

  10. [17]

    Y. Li. Existence of many positive solutions of semilinear elliptic equations on annulus. J. Differential Equations, 83(2):348–367, 1990

    1990

  11. [19]

    C. Lin. Uniqueness of least energy solutions to a semilinear elliptic equation inR 2. Manuscripta Math., 84(1):13–19, 1994

    1994

  12. [20]

    A. I. Nazarov. The one-dimensional character of an extremum point of the Friedrichs inequality in spherical and plane layers.J. Math. Sci., 102(5):4473–4486, 2000

    2000

  13. [21]

    H. Zou. On the effect of the domain geometry on uniqueness of positive solutions of ∆u+u p = 0.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21(3):343–356, 1994. Address and E-mail: Rongxun He School of Mathematical Sciences, Fudan University, Shanghai 200433, China rxhe24@m.fudan...

    1994

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