Pith. sign in

REVIEW 9 minor 25 references

Supercritical Lane-Emden equation on a cone with an inhomogeneous Dirichlet boundary condition

T0 review · 0 major / 9 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single cone eigenvalue sets the threshold structure of supercritical Lane–Emden solutions.

desk verdict Solid, genuinely new classification result for the Lane-Emden equation on cones; the domain-dependent exponent condition is real, and the paper deserves a serious referee. read the letter →

arxiv 2411.14686 v1 pith:DERHQPD7 submitted 2024-11-22 math.AP

Sho Katayama This is my paper · ORCID
classification math.AP MSC 35J2535J6135B0935B32
keywords Lane–EmdenequationsupercriticalnonlinearityinfiniteconeinhomogeneousDirichletboundaryconditionJoseph–LundgrenexponentmultiplepositivesolutionsHardyinequalitybifurcationtheory
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 studies the Lane–Emden equation $-\Delta u=u^p$ on an infinite cone in $\mathbb{R}^N$, with a prescribed boundary value $\kappa\mu$ on the cone's boundary away from the tip. It asks how the size $\kappa$ of the boundary data orders the set of positive solutions, and answers with a classification in the supercritical range $p>p^*_\gamma$, $p\kappa^*$, and at least two solutions exist for $\kappa_*<\kappa<\kappa^*$. The decisive object is the cubic $H(q)$ of (1.12), whose coefficients involve only the dimension $N$ and the first eigenvalue $\Lambda$ of the cone's cross-section, so the shape of the domain enters the boundary-data dichotomy through a single spectral quantity. The paper also proves a threshold-existence theorem (Theorem 1.1) valid for every $p>p^*_\gamma$, and extends both results to Hénon-type nonlinearities $K(x)u^p$ with $K(x)\sim |x|^a$.

What carries the argument

Three mechanisms together carry the argument. The weighted space $C_{\alpha,\beta}$ with weight $U_{\alpha,\beta}(x)=|x|^\alpha(1+|x|^2)^{(\beta-\alpha)/2}$ encodes admissible growth at the tip and decay at infinity, and the nonlinear problem is rewritten as a fixed-point equation $\Phi(u,\kappa)=0$ with $\Phi$ a smooth map on this Banach space. The sharp conical Hardy inequality $((N-2)/2)^2+\Lambda)\int_\Omega |x|^{-2}\varphi^2\le \int_\Omega |\nabla\varphi|^2$ supplies the coercivity behind the new weighted energy estimates (Lemmas 5.3 and 5.4), and a direct calculation converts the resulting condition (5.32) into the cubic inequality $H(p-1)<0$. Finally, the linearized operator at the extremal solution has one-dimensional kernel spanned by the first eigenfunction $\varphi_{\kappa^*}$, and a standard bifurcation theorem (the paper's reference [7, Theorem 3.2]) turns that nondegeneracy into a curve of solutions bending back from $\kappa^*$, producing the second branch.

What would settle it

Fix a cone with cross-section $A$ where $H(p-1)<0$ and $p<p_{JL}$, and numerically continue the branch of positive solutions of $(L_\kappa)$ as $\kappa$ increases: the theorem predicts a minimal branch up to $\kappa_*$, a second branch for $\kappa_*<\kappa<\kappa^*$, a single solution at $\kappa^*$, and none above it. A branch count different from that — for instance, no solution at $\kappa^*$, three solutions in the interval, or a solution for $\kappa>\kappa^*$ — would refute the classification. Separately, checking the best constant in (1.16) for that cone decides whether the external Hardy-inequality input really is sharp.

Watch

Extended reading notes

Core claim

On its own terms, the paper proves that whenever $p>p^*_\gamma$, $p<p_{JL}$, and $H(p-1)<0$, the solution set of $(L_\kappa)$ is organized by two constants $0\le\kappa_*<\kappa^*$: the admissible set $K=\{\kappa>0: (L_\kappa)\text{ has a solution}\}$ is exactly $(0,\kappa^*]$, with a minimal solution on $0<\kappa\le\kappa_*$, a unique solution at $\kappa=\kappa^*$, no solutions for $\kappa>\kappa^*$, and at least two solutions for $\kappa_*<\kappa<\kappa^*$, one of them pointwise above the minimal one. The proof builds a supersolution for small $\kappa$, shows that minimal solutions are stable, uses new weighted energy estimates to control their decay, passes to the limit to attain $\kappa^*$, and then applies a bifurcation theorem whose transversality at $\kappa^*$ forces a backward-folding branch. The boundary data $\mu$ enter only through growth conditions at the tip and at infinity; the condition $H(p-1)<0$ is exactly the coercivity condition for the weighted estimates once the cone's Hardy constant $((N-2)/2)^2+\Lambda$ is inserted.

Load-bearing premise

The load-bearing premise is that the cone Hardy inequality with the sharp constant $((N-2)/2)^2+\Lambda$, cited from [25], is valid on $\Omega$; this is the exact inequality whose coercivity condition later becomes $H(p-1)<0$, so if the constant were smaller or the inequality failed, the decay estimates behind critical existence and multiplicity would collapse.

Editorial extensions

If this is right

  • For any cone whose first eigenvalue $\Lambda$ is known, the full two-threshold picture is valid on the exponent interval $p_1<p<p_{JL}$, where $p_1>p^*_\gamma$ is the root of $H(p-1)=0$; outside this interval only the existence/nonexistence threshold of Theorem 1.1 is asserted.
  • At $\kappa=\kappa^*$, the unique solution is the limit of the minimal solutions $u_\kappa$ as $\kappa\uparrow\kappa^*$, so the minimal branch extends continuously to the endpoint.
  • For $\kappa_*<\kappa<\kappa^*$, the bifurcation branch yields two distinct positive solutions, one strictly larger than the minimal one, so the minimal solution does not exhaust the solution set in that interval.
  • The same classification holds for Hénon-type equations $-\Delta u=K(x)u^p$ with $K(x)\sim |x|^a$, with the cubic replaced by $H_a(p-1)$; in some ranges of $a$ the admissible exponents form a union of two intervals (Theorem 7.2).

Reading between the lines

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

  • The paper leaves the cases $p\ge p_{JL}$ and $H(p-1)\ge0$ open. A natural test is to continue the branch numerically past $\kappa^*$ in one of those regimes: the outcome would show whether the failure of $H(p-1)<0$ corresponds to the endpoint not being attained, the branch not folding, or a different multiplicity pattern; the paper does not predict which.
  • Because the same cubic $H(p-1)=0$ appears in a related stability/Liouville analysis cited in the paper, the turning point $\kappa^*$ may coincide with the point where the minimal solution loses linearized stability; the paper does not draw this connection, but its eigenvalue computation at $\kappa^*$ provides a concrete object to test it.
  • Since the proof's coercivity depends on the cone's Hardy constant, one would expect the same classification on any domain for which the Hardy constant is $((N-2)/2)^2+\Lambda$; comparing cones with the same cross-section spectrum but different geometry would show how much of the phenomenon is spectral and how much depends on finer shape.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 9 minor

Summary. The paper studies the supercritical Lane–Emden equation -Δu = u^p on an infinite cone Ω with inhomogeneous Dirichlet data κμ on ∂Ω\{0}. The main results are Theorem 1.1, which gives a threshold κ* such that minimal solutions exist for 0<κ<κ* and no solutions exist for κ>κ*, and Theorem 1.2, which under p<p_JL and H(p-1)<0 (with H the cubic in (1.12) depending on N and the cone eigenvalue Λ) adds existence of a unique solution at κ=κ* and the existence of at least two solutions for κ in an interval (κ*,κ*). The proofs combine a Perron/supersolution construction, linearized eigenvalue problems and stability of minimal solutions, new weighted energy estimates based on the sharp conical Hardy inequality (1.16), and a Crandall–Rabinowitz bifurcation argument. Section 7 states analogous results for Hénon-type equations. The algebraic passage from (5.32) to H(p-1)<0 is correct, and the condition H(p-1)<0 is derived rather than imposed.

Significance. If the results are correct, the paper gives a rather complete existence/nonexistence and multiplicity classification for a supercritical boundary-value problem on cones, extending to the critical boundary case and to multiplicity earlier work on bounded domains, the half-space, and the forced Lane–Emden equation. The main novelty is a set of weighted energy estimates whose coercivity condition is linked to the constant ((N-2)/2)^2+Λ in the sharp Hardy inequality on cones; this produces a genuinely domain-dependent cubic condition H(p-1)<0. The proof structure is coherent, and the paper is careful to distinguish what is proved from what is imported from the literature. In particular, no free parameters are fitted and the threshold condition is not assumed ad hoc. The main caveat is that the pivotal estimate (5.21) relies on the cited sharp Hardy inequality (1.16), so the authors should make the applicability of that inequality explicit; with that clarification, the argument is convincing.

minor comments (9)
  1. [Section 3, proof of Proposition 3.1] The displayed inequalities 'α−2>pα and β−2<pβ' have the wrong direction. From (1.8) one has α−2<pα and β−2>pβ, which are precisely the inequalities needed for the lower bound −ΔV_{α,β}≥CU_{pα,pβ}. Please correct the direction of these inequalities.
  2. [Section 5, Lemma 5.3, Eq. (5.23)] In the second inequality of (5.23), the term τ²|x|^{2τ−2}(wκ)^2 should carry the cutoff ψ², i.e. it should read τ²ψ²|x|^{2τ−2}(wκ)^2. Without ψ², the subsequent application of the Hardy inequality (5.30) to ψ|x|^τ wκ is not justified, and the displayed bound is not what is used later.
  3. [Section 5, Lemma 5.3, proof] The sentence 'Combining (5.23)–(5.26), (5.30), and (5.31)' appears to contain a wrong cross-reference: (5.31) is the conclusion of the lemma rather than an input. It should refer to (5.29) in place of (5.31).
  4. [Section 5, Lemma 5.4, proof] The text 'We find τ1 and τ2 satisfying (5.2)' should refer to condition (5.20) (or the displayed system (5.34)) rather than to (5.2), which is a different condition in Lemma 5.1.
  5. [Section 5, Lemma 5.1, Eqs. (5.17)–(5.18)] In (5.17) and (5.18), the exponent inside the integral appears to be 2qν/(p−1), not 2q/((p−1)ν), if the displayed estimate is to follow from Lemma 5.1 with the outer exponent (p−1)/(2qν). Please correct the exponent and verify the surrounding algebra.
  6. [Section 5, Lemma 5.3, Eq. (5.30)] Please state explicitly that ψ|x|^τ wκ belongs to D^{1,2}_0(Ω), for example via Lemma 4.4(iii) and the compact support of ψ away from the vertex and infinity, so that the conical Hardy inequality (1.16) applies. It would also be helpful to quote the precise statement of [25, Proposition 4.1] to confirm that the constant A=((N-2)/2)^2+Λ is exactly the one used.
  7. [Section 4 vs. Section 5] The symbol vκ is introduced in Section 4 as vκ=vκ_{j*}, but in the estimates around (5.7) the relevant quantity appears to be vκ_{j*+1}. Please reconcile the notation or explicitly state which index is used in Lemma 5.1 and in (5.26).
  8. [Proposition 6.1] The statement of the Crandall–Rabinowitz theorem includes both Fκ(u*,κ*)∉Im(Fu) and Fκ(u*,κ*)∉Z, where Z is merely a complementary subspace of the kernel. In the standard theorem these are not independent hypotheses; the correct assumption is non-membership in the image. Since in the application Z is taken to be the image, please rephrase the proposition to avoid this ambiguity.
  9. [Section 7, Theorem 7.2] The opening sentence 'Assume the same conditions as in Theorem 1.1' should presumably read 'Theorem 7.1' when stating the Hénon-type generalization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.2's threshold H(p−1)<0 is derived from an external conical Hardy inequality and a direct algebraic calculation, not fitted or imported from the paper's own conclusions.

full rationale

The derivation chain is self-contained in the relevant sense. The central condition H(p−1)<0 is not imposed as an input but emerges from the proofs: Lemma 5.3 requires the weighted coercivity condition (5.20), Lemma 5.4 uses the resulting threshold (5.32), and the proof of Proposition 5.1 verifies by direct calculation that (5.32) is algebraically equivalent to H(p−1)<0. This is a genuine derivation, not a restart of the theorem's conclusion. The only external geometric input is the sharp conical Hardy inequality (1.16) with constant ((N−2)/2)^2+Λ, cited to [25, Proposition 4.1]. That result concerns the first Laplace–Beltrami eigenvalue Λ on the spherical domain A and is an independent published analytic fact; it is not equivalent to the Lane–Emden classification, and the paper explicitly identifies it as the source of the domain's influence on the threshold. The self-citations and prior-paper lemmas used in the proofs, such as [17, Lemma 2.3] and [18, Lemmas 4.6 and 5.1], supply auxiliary comparison, compactness, and energy estimates; none states Theorem 1.2 or smuggles in the H(p−1)<0 condition. The bifurcation argument uses the external Crandall–Rabinowitz theorem. If the cited Hardy constant were incorrect, Theorems 1.1–1.2 could lose support, but that would be an external correctness risk, not circularity. No fitted parameter is relabeled as a prediction, no equation is defined in terms of the target result, and no self-citation chain forces the conclusion. Accordingly, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No new physical or abstract objects are postulated. The paper introduces only a cubic polynomial H and the weighted norms C^{alpha,beta}; these are mathematical constructions, not entities with independent falsifiable handles. The analysis uses standard PDE machinery and prior results, not new invented mechanisms.

assumptions (6)
  • domain assumption Sharp Hardy inequality on cones: (((N-2)/2)^2 + Lambda) * integral |x|^{-2} phi^2 <= integral |grad phi|^2 for phi in D^{1,2}_0(Omega)
    Invoked as (1.16) from Nazarov [25, Proposition 4.1]; the constant appears in the weighted estimates (5.20), (5.30), and (5.32) that produce the condition H(p-1)<0.
  • standard math Joseph-Lundgren exponent construction: existence of nu >= 1 and q satisfying (5.2) and (5.16) for 1 < p < p_JL
    Used in Lemma 5.2 to obtain fast decay of solutions; the proof refers to [18, Lemma 5.4] for the existence of such nu and q.
  • standard math Linear theory for weighted Poisson problems on cones: Proposition 2.1 (existence, uniqueness, estimates) and Proposition 2.2 (compactness, D^{1,2}_0 membership)
    Proved in Section 2 via Perron's method, Ascoli-Arzela, Lax-Milgram, and standard elliptic regularity from [16]. This is the backbone of the Green operator G used throughout.
  • standard math Maximum principle and Perron method on cones for the weighted classes C^{alpha,beta}
    Used in Lemmas 2.1, 2.2, and Proposition 2.1 to construct barriers and unique solutions for the linearized problems.
  • domain assumption Berestycki-Capuzzo-Dolcetta-Nirenberg Liouville theorem: for p <= p*_gamma there is no positive supersolution to -Delta u >= u^p on a cone
    Sets the threshold p > p*_gamma in Theorems 1.1 and 1.2; cited as [4] and taken as known background for the necessary condition.
  • domain assumption Eigenvalue problem (E_kappa) admits a unique first eigenpair with positivity, simplicity, and the stability inequality (Lemma 4.1)
    Proof deferred to a similar argument in [24, Lemma B.2]. This is load-bearing for the stability of minimal solutions (Proposition 4.1).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Supercritical Lane-Emden equation on a cone with an inhomogeneous Dirichlet boundary condition." pith.science (2026). https://pith.science/paper/DERHQPD7

@misc{pith2026241114686,
  author       = {Pith},
  title        = {Pith review of: Supercritical Lane-Emden equation on a cone with an inhomogeneous Dirichlet boundary condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DERHQPD7}},
  note         = {Machine review of arXiv:2411.14686}
}
read the original abstract

We consider the Lane-Emden equation with a supercritical nonlinearity with an inhomogeneous Dirichlet boundary condition on an infinite cone. Under suitable conditions for the boundary data and the exponent of nonlinearity, we give a complete classification of the existence/nonexistence of a solution with respect to the size of boundary data. Moreover, we give a result on the multiple existence of solutions via bifurcation theory. We also state results on Hardy-H\'enon equations on infinite cones as a generalization.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Ai and X

    J. Ai and X. P. Zhu, Positive solutions of inhomogeneous elliptic boundary val ue problems in the half space , Comm. Partial Differential Equations 15 (1990), 1421–1446

    work page 1990

  2. [2]

    Bandle and M

    C. Bandle and M. Ess´ en, On positive solutions of Emden equations in cone-like domai ns, Arch. Rational Mech. Anal. 112 (1990), 319–338

    work page 1990

  3. [3]

    Bidaut-V´ eron, G

    M.-F. Bidaut-V´ eron, G. Hoang, Q.-H. Nguyen, and L. V´ er on, An elliptic semilinear equation with source term and boundary measure data: the supercritical case , J. Funct. Anal. 269 (2015), 1995–2017

    work page 2015

  4. [4]

    Berestycki, I

    H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems , Topol. Methods Nonlinear Anal. 4 (1994), 59–78

    work page 1994

  5. [5]

    Bidaut-V´ eron and S

    M.-F. Bidaut-V´ eron and S. Pohozaev,Nonexistence results and estimates for some nonlinear elli ptic problems, J. Anal. Math. (2001), 1–49

    work page 2001

  6. [6]

    Bidaut-V´ eron and L

    M.-F. Bidaut-V´ eron and L. Vivier, An elliptic semilinear equation with source term involving boundary mea- sures: the subcritical case , Rev. Mat. Iberoamericana 16 (2000), 477–513

    work page 2000

  7. [7]

    M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linea rized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180

    work page 1973

  8. [8]

    Dupaigne, A

    L. Dupaigne, A. Farina, and T. Petitt, Liouville-type theorems for the Lane-Emden equation in the half-space and cones , J. Funct. Anal. 284 (2023), Paper No. 109906, 27

    work page 2023

Show all 25 references
  1. [9]

    A. J. Fern´ andez and T. Weth, The nonlinear Schr¨ odinger equation in the half-space, Math. Ann. 383 (2022), 361–397

    2022

  2. [10]

    M. Fila, K. Ishige, and T. Kawakami, Large-time behavior of solutions of a semilinear elliptic e quation with a dynamical boundary condition , Adv. Differential Equations 18 (2013), 69–100. 28

    2013

  3. [11]

    M. Fila, K. Ishige, and T. Kawakami, Large-time behavior of small solutions of a two-dimensiona l semilinear elliptic equation with a dynamical boundary condition , Asymptot. Anal. 85 (2013), 107–123

    2013

  4. [12]

    , Existence of positive solutions of a semilinear elliptic eq uation with a dynamical boundary condition , Calc. Var. Partial Differential Equations 54 (2015), 2059–2078

    2015

  5. [13]

    Elliptic Parabol

    , Positive solutions of a semilinear elliptic equation with s ingular Dirichlet boundary data , J. Elliptic Parabol. Equ. 1 (2015), 335–362

    2015

  6. [14]

    , Minimal solutions of a semilinear elliptic equation with a d ynamical boundary condition , J. Math. Pures Appl. (9) 105 (2016), 788–809 (English, with English and French summarie s)

    2016

  7. [15]

    Gidas and J

    B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinea r elliptic equations , Comm. Pure Appl. Math. 34 (1981), 525–598

    1981

  8. [16]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order , Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition

    2001

  9. [17]

    Ishige and S

    K. Ishige and S. Katayama, Supercritical H´ enon-type equation with a forcing term, Adv. Nonlinear Anal. 13 (2024), Paper No. 20240003, 28

    2024

  10. [18]

    Ishige, S

    K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term , J. Math. Pures Appl. 128 (2019), 183–212

    2019

  11. [19]

    N. J. Kalton and I. E. Verbitsky, Nonlinear equations and weighted norm inequalities , Trans. Amer. Math. Soc. 351 (1999), 3441–3497

    1999

  12. [20]

    Katayama, Semilinear elliptic problems on the half space with a superc ritical nonlinearity, Discrete Contin

    S. Katayama, Semilinear elliptic problems on the half space with a superc ritical nonlinearity, Discrete Contin. Dyn. Syst. 44 (2024), 3774–3806

    2024

  13. [21]

    G. G. Laptev, Absence of global positive solutions of systems of semiline ar elliptic inequalities in cones , Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 107–124 (Russian, with Russian summary); English t ransl., Izv. Math. 64 (2000), 1197–1215

    2000

  14. [22]

    Li and G

    Y. Li and G. Xie, Positive solutions for semilinear elliptic systems with bo undary measure data , Ann. Mat. Pura Appl. (4) 201 (2022), 1325–1346

    2022

  15. [23]

    Mitidieri and S

    `E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities , Tr. Mat. Inst. Steklova 234 (2001), 1–384 (Russian, with English and Russian summaries); English transl., Proc. Steklov Inst. M ath...

    2001

  16. [24]

    Naito and T

    Y. Naito and T. Sato, Positive solutions for semilinear elliptic equations with singular forcing terms , J. Differential Equations 235 (2007), 439–483

    2007

  17. [25]

    A. I. Nazarov, Hardy-Sobolev inequalities in a cone , J. Math. Sci. (N.Y.) 132 (2006), 419–427. Problems in mathematical analysis. No. 31. 29

    2006

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.