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Liouville theorems for $p$-Laplacian equations in convex cones without finite-energy condition

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Pith's one-line read Every bounded nonnegative solution to the subcritical anisotropic p-Laplacian equation vanishes in convex cones.

desk verdict Extends subcritical Liouville results and critical classification to convex cones while dropping the finite-energy assumption for p in ((N+1)/3, N). read the letter →

arxiv 2605.29281 v1 pith:XAAI52FZ submitted 2026-05-28 math.AP math.DG

Lu Chen , Wei Dai , Changfeng Gui , Yunpeng Luo This is my paper · ORCID
classification math.APmath.DG
keywords Liouvilletheoremp-LaplacianconvexconeanisotropicoperatorsubcriticalgrowthNeumannboundaryblow-upmethod
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 proves Liouville theorems for the anisotropic Finsler p-Laplacian equation in open convex cones with Neumann boundary conditions. If the right-hand side f(u) is nonnegative and subcritical, every bounded nonnegative solution must be identically zero. For subcritical power nonlinearities, a pointwise decay estimate is established using the doubling argument and blowing-up method, showing that all nonnegative solutions vanish even without the boundedness assumption. These results extend previous theorems from whole space to cones and complement the critical case classification.

What carries the argument

The anisotropic Finsler p-Laplacian operator with homogeneous Neumann boundary condition on the boundary of the convex cone.

What would settle it

Constructing a nonzero bounded nonnegative solution to the equation in some open convex cone would disprove the main claim.

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Extended reading notes

Core claim

If f(u) is nonnegative and subcritical, every bounded nonnegative solution in C is identically zero. In particular, for f(u)=u^q with 0<q<p^*-1, all nonnegative solutions must be zero without the boundedness assumption. For the critical case f(u)=u^{p^*-1} and H(xi)=|xi|, positive solutions are classified for (N+1)/3 < p < N without finite-energy assumption.

Load-bearing premise

The domain is an open convex cone and the nonlinearity satisfies the subcritical growth condition.

Editorial extensions

If this is right

  • Nonnegative solutions to subcritical power equations are identically zero in the cone.
  • Pointwise decay estimates hold for subcritical solutions.
  • Positive solutions in the critical case are classified for p in ((N+1)/3, N) without finite energy.
  • Results extend from Euclidean space to general convex cones.

Reading between the lines

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

  • Similar vanishing results might hold for other anisotropic operators in cones.
  • The classification in the critical case could extend to more general H.
  • These zero conclusions may imply nonexistence for related parabolic problems in cones.
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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

0 major / 3 minor

Summary. The paper establishes Liouville theorems for the anisotropic Finsler p-Laplacian equation with Neumann boundary conditions in open convex cones C. For nonnegative subcritical f, every bounded nonnegative solution is identically zero. For f(u)=u^q with 0<q<p^*-1, a pointwise decay estimate is obtained via doubling and blow-up arguments, implying all nonnegative solutions vanish (without a boundedness assumption). In the critical case f(u)=u^{p^*-1} with H(ξ)=|ξ|, positive solutions are classified for (N+1)/3 < p < N without a finite-energy assumption. These extend results from R^N (SZ, CFV, CHN, Ou) and prior cone results (CFR) to general convex cones.

Significance. If the proofs hold, the results meaningfully extend Liouville-type theorems to convex cones while removing the finite-energy hypothesis in the critical case for the standard p-Laplacian. The doubling/blow-up approach is standard in the cited literature but applied here to a new geometric setting with the natural Neumann condition; this is a solid incremental contribution to the theory of p-Laplacian equations in non-Euclidean domains.

minor comments (3)
  1. [Abstract] Abstract: the critical Sobolev exponent p^* is used without an explicit definition (standardly p^* = Np/(N-p)); add a brief parenthetical for clarity.
  2. [Abstract] Abstract, critical-case paragraph: the restriction to H(ξ)=|ξ| and the range (N+1)/3 < p < N should be cross-referenced to the corresponding hypothesis in Ou to make the extension transparent.
  3. [Introduction] The manuscript should confirm in the introduction or methods section that the convexity of C is used only to preserve the cone structure under rescaling and that the Neumann condition passes to the limit without additional assumptions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment. We appreciate the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The derivation applies doubling arguments and blow-up methods to obtain decay estimates and zero conclusions for subcritical f(u) in convex cones C, then extends critical-case classification for H(ξ)=|ξ| when (N+1)/3 < p < N. These steps rely on the stated PDE, Neumann boundary condition, and geometric assumptions on C, together with standard techniques from the cited external literature (SZ, CFV, CHN, CFR, Ou). No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the proofs are logically independent of the target results.

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

The central claims rest on domain assumptions about the cone and growth conditions on f; no free parameters or invented entities are introduced in the abstract.

assumptions (2)
  • domain assumption C is an open convex cone in R^N with N >= 3
    Invoked in the statement of the PDE and all results; convexity is used for the boundary condition and symmetry arguments.
  • domain assumption f(u) is nonnegative and subcritical (0 < q < p^* - 1 for power case)
    Required for the zero conclusion and decay estimate via doubling and blow-up.

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Pith. "Pith review of Liouville theorems for $p$-Laplacian equations in convex cones without finite-energy condition." pith.science (2026). https://pith.science/paper/XAAI52FZ

@misc{pith2026260529281,
  author       = {Pith},
  title        = {Pith review of: Liouville theorems for $p$-Laplacian equations in convex cones without finite-energy condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XAAI52FZ}},
  note         = {Machine review of arXiv:2605.29281}
}
abstract

We study the anisotropic Finsler $p$-Laplacian equation \begin{equation*} \left\{ \begin{aligned} &-\Delta ^{H}_{p}u=f(u) \quad\,\,\, &{\rm{in}} \,\, \mathcal{C}, &{\bf{a}}(\nabla u)\cdot \nu =0 \quad\,\,\, &{\rm{on}} \,\, \partial\mathcal{C}, \end{aligned} \right. \end{equation*} where $N\geq3$, $1<p<N$, $\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone and $\Delta ^{H}_{p}$ is the anisotropic Finsler $p$-Laplacian operator. If $f(u)$ is nonnegative and subcritical, we prove that every bounded nonnegative solution in $\mathcal{C}$ is identically zero. In particular, for $f(u)=u^{q}$ with $0<q<p^*-1$, we establish a pointwise decay estimate in $\mathcal{C}$ via the doubling argument and blowing-up method and prove that all nonnegative solutions must be zero without the boundedness assumption. Our results are the subcritical counterpart of the classification result for the critical case in \cite{CFR}, and extend the Liouville type theorems in $\mathbb{R}^{N}$ for the standard $p$-Laplacian in \cite{SZ} and for the anisotropic $p$-Laplacian in \cite{CFV, CHN} to general convex cones $\mathcal{C}$. In the critical case $f(u)=u^{p^*-1} $ and typical case $H(\xi)=|\xi|$, for $\frac{N+1}{3}<p<N$, we classified the positive solutions of the critical $p$-Laplacian equation in convex cones $\mathcal{C}$ without finite-energy assumption. This extends the classification result of \cite{Ou} in $\mathbb{R}^{N}$ to general convex cones $\mathcal{C}$, and removes the finite-energy assumption in \cite{CFR} in the typical case $H(\xi)=|\xi|$.

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Reference graph

Works this paper leans on

73 extracted references · 8 canonical work pages · cited by 3 Pith papers

  1. [1]

    Acerbi and N

    E. Acerbi and N. Fusco,Regularity for minimizers of non-quadratic functionals: the case1< p <2, J. Math. Anal. Appl.,140(1989), 115-135

    1989

  2. [2]

    C. A. Antonini, G. Ciraolo and A. Farina,Interior regularity results for inhomogeneous anisotropic quasilinear equations, Math. Ann.,2023(387), 1745-1776

    2023

  3. [3]

    Avelin, T

    B. Avelin, T. Kuusi and G. Mingione,Nonlinear Calder´ on-Zygmund theory in the limiting case, Arch. Ration. Mech. Anal.,227(2018), 663-714

    2018

  4. [4]

    Belloni, V

    M. Belloni, V. Ferone and B. Kawohl,Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Special issue dedicated to Lawrence E. Payne, Z. Angew. Math. Phys.,54(2003), 771-783

    2003

  5. [5]

    Bianchini and G

    C. Bianchini and G. Ciraolo,Wulff shape characterizations in overdetermined anisotropic elliptic problems, Comm. Partial Differential Equations,43(2018), 790-820

    2018

  6. [6]

    Bidaut-Veron,Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch

    M.-F. Bidaut-Veron,Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal.,107(1989), 293-324

    1989

  7. [7]

    Brezis,Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011, xiv+599 pp

    H. Brezis,Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011, xiv+599 pp

    2011

  8. [8]

    Cabr´ e, X

    X. Cabr´ e, X. Ros-Oton and J. Serra,Sharp isoperimetric inequalities via the ABP method, J. Eur. Math. Soc., 18(2016), 2971-2998

    2016

Show all 73 references
  1. [9]

    Caffarelli, N

    L. Caffarelli, N. Garofalo and F. Segala,A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math.,47(1994), 1457-1473

    1994

  2. [10]

    Caffarelli, B

    L. Caffarelli, B. Gidas, J. Spruck,Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math.,42(1989), 271-297

    1989

  3. [11]

    D. Cao, W. Dai and Y. Li,Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to weighted doublyD 1,p-critical quasi-linear nonlocal elliptic equations with Hardy potential, Sci. China Math., 69(2026), 1143-1196

    2026

  4. [12]

    Catino, D

    G. Catino, D. Monticelli and A. Roncoroni,On the criticalp-laplace equation, Adv. Math.,433(2023), Paper No. 109331, 38 pp

    2023

  5. [13]

    S.-Y. A. Chang and P. C. Yang,On uniqueness of solutions ofn-th order differential equations in conformal geometry, Math. Res. Lett.,4(1997), 91-102

    1997

  6. [14]

    Chanillo and M

    S. Chanillo and M. K.-H. Kiessling,Conformally invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal.,5(1995), no. 6, 924-947

    1995

  7. [15]

    L. Chen, T. Wu, J. Yan and Y. Yang,Classification of positive solutions of critical anisotropic Sobolev equation without the finite volume constraint, arXiv:2404.11628, 2025

    2025

  8. [16]

    Cheng, G

    T. Cheng, G. Huang and Y. Niu,Classification of solutions to Finslerp-Laplacian equation, Discrete Contin. Dyn. Syst.,48(2026), 577-629

    2026

  9. [17]

    Chen and C

    W. Chen and C. Li,Classification of solutions of some nonlinear elliptic equations, Duke Math. J.,62(1991), no. 3, 615-622

    1991

  10. [18]

    Chou and T

    K. Chou and T. Y. H. Wan,Asymptotic radial symmetry for solutions of∆u+e u = 0in a punctured disc, Pac. J. Math.,163(1994), 269-276

    1994

  11. [19]

    Cianchi and V

    A. Cianchi and V. Maz’ya,Second-Order Two-Sided Estimates in Nonlinear Elliptic Problems, Arch. Rational Mech. Anal.,229(2018), 569-599

    2018

  12. [20]

    Ciraolo, A

    G. Ciraolo, A. Figalli and A. Roncoroni,Symmetry results for critical anisotropicp-Laplacian equations in convex cones, Geom. Funct. Anal.,30(2020), 770-803

    2020

  13. [21]

    Ciraolo and X

    G. Ciraolo and X. Li,An exterior overdetermined problem for FinslerN-Laplacian in convex cones, Calc. Var. Partial Differential Equations,61(2022), Paper No. 121, 27 pp

    2022

  14. [22]

    Ciraolo and X

    G. Ciraolo and X. Li,Classification of solutions to the anisotropicN-Liouville equation inR N, Int. Math. Res. Not.,19(2024), 12824-12856

    2024

  15. [23]

    Cozzi, A

    M. Cozzi, A. Farina and E. Valdinoci,Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Adv. Math.,293(2016), 343-381

    2016

  16. [24]

    W. Dai, L. Duan, C. Gui and Y. Li,Non-radial solutions for the critical quasi-linear H´ enon equation involving p-Laplacian inR N, Proc. London Math. Soc.,132(2026), no. 4, Paper No. e70148, 61 pp

    2026

  17. [25]

    W. Dai, C. Gui, Y. Hu and S. Peng,Nonlinear Neumann boundary problems forn-Laplacian Liouville equation on a half space, arXiv:2602.06414, 2026, 49 pp

    2026

  18. [26]

    W. Dai, C. Gui and Y. Luo,Anisotropic FinslerN-Laplacian Liouville Equation in Convex Cones, arXiv:2407.04987, 2024, 38 pp. 38 LU CHEN, WEI DAI, CHANGFENG GUI, YUNPENG LUO

    2024

  19. [27]

    W. Dai, Y. Li and Z. Liu,Radial symmetry and sharp asymptotic behaviors of nonnegative solutions toD 1,p- critical quasi-linear static Schr¨ odinger-Hartree equation involvingp-Laplacian−∆p, Math. Ann.,391(2025), 2653-2708

    2025

  20. [28]

    W. Dai, Z. Liu and B. Sciunzi,Classification of solutions to−∆u=e −2u in the half-space, Rev. Mat. Iberoam., 42(2026), no. 3, 1039-1058

    2026

  21. [29]

    Dai and G

    W. Dai and G. Qin,Classification of nonnegative classical solutions to third-order equations, Adv. Math.,328 (2018), 822-857

    2018

  22. [30]

    Dai and G

    W. Dai and G. Qin,Liouville type theorems for fractional and higher order H´ enon-Hardy type equations via the method of scaling spheres, Int. Math. Res. Not. IMRN,2023(2023), no. 11, 9001-9070

    2023

  23. [31]

    Dai and G

    W. Dai and G. Qin,Method of scaling spheres: Liouville theorems in general bounded or unbounded domains, blowing-up analysis on not necessarilyC 1-smooth domains and other applications, 2023, preprint, submitted for publication, arXiv: 2302.13988, 78 pp

    2023

  24. [32]

    Dacorogna and C

    B. Dacorogna and C. E. Pfister,Wulff theorem and best constant in Sobolev inequality, J. Math. Pures Appl., 71(1992), 97-118

    1992

  25. [33]

    Damascelli,Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann

    L. Damascelli,Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire,15(4): 493-516, 1998

    1998

  26. [34]

    Damascelli, A

    L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci,Liouville results form-Laplace equations of Lane- Emden-Fowler type, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire,26(2009), no. 4, 1099-1119

    2009

  27. [35]

    Damascelli, S

    L. Damascelli, S. Merch´ an, L. Montoro and B. Sciunzi,Radial symmetry and applications for a problem involving the−∆ p(·)operator and critical nonlinearity inR N, Advances in Mathematics,265(2014), 313- 335

    2014

  28. [36]

    DiBenedetto,C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.,7 (1983), 827-850

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

    1983

  29. [37]

    Dipierro, G

    S. Dipierro, G. Poggesi and E. Valdinoci,Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities, Calc. Var. Partial Differential Equations,61(2022), Paper No. 72, 31 pp

    2022

  30. [38]

    J. Dou, Y. Han, S. Yuan and Y. Zhou,Sharp Onofri trace inequality on the upper half space and quasi-linear Liouville equation with Neumann boundary, arXiv:2509.17031

  31. [39]

    J. Dou, T. Wu and S. Yuan,A sharp anisotropic Onofri inequality and Liouville theorem for anisotropic p-Laplacian equations inR N, 2026, 25 pp

    2026

  32. [40]

    L. C. Evans,A new proof of localC 1,α regularity for solutions of certain degenerate elliptic P. D. E., J. Differential equations,45(1982), 525-598

    1982

  33. [41]

    Fonseca and N

    I. Fonseca and N. Fusco,Regularity results for anisotropic image segmentation models, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV),24(1997), 463-499

    1997

  34. [42]

    Fonseca and S

    L. Fonseca and S. M¨ uller,A uniqueness proof of Wulff Theorem, Proc. Royal Soc. Edinburgh - A: Math.,119 (1991), 125-136

    1991

  35. [43]

    Gidas, W

    B. Gidas, W. M. Ni and L. Nirenberg,Symmetry and related properties via the maximum principle, Comm. Math Phys.,68(1979), 209-243

    1979

  36. [44]

    Gidas and J

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

    1981

  37. [45]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger,Elliptic partial differential equations of second order, Springer-Verlag, Berlin- New York (1977)

    1977

  38. [46]

    Gui and A

    C. Gui and A. Moradifam,The sphere covering inequality and its applications, Invent. Math.,214(2018), no. 3, 1169-1204

    2018

  39. [47]

    J. L. Lewis,Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J.,32(1983), 849-858

    1983

  40. [48]

    Liang, T

    W. Liang, T. Wu and J. Yan,Liouville theorems for anisotropicp-Laplacian equations with a semilinear term, arXiv:2507.20182v1, 2025

    2025

  41. [49]

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

    1988

  42. [50]

    C. S. Lin,A classification of solutions of a conformally invariant fourth order equation inR n, Comment. Math. Helv.,73(1998), 206-231

    1998

  43. [51]

    Ma and Q

    X. Ma and Q. Ou,A Liouville theorem for a class semilinear elliptic equations on the Heisenberg group, Adv. Math.,413(2023), Paper No. 108851, 20 pp

    2023

  44. [52]

    X. Ma, Q. Ou and T. Wu,Jerison-Lee identities and Semi-linear subelliptic equations on CR manifolds, preprint, arXiv:2311.16428, 38 pp

  45. [53]

    Q. Z. Ou,On the classification of entire solutions to the criticalp-Laplace equation, Math. Ann.,392(2025), no. 2, 1711-1729. LIOUVILLE THEOREMS FORp-LAPLACIAN EQUATIONS IN CONVEX CONES 39

    2025

  46. [54]

    L. A. Peletier and J. Serrin,Gradient bounds and Liouville theorems for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4),5(1978), 65-104

    1978

  47. [55]

    Peral,Multiplicity of solutions for thep-Laplacian

    I. Peral,Multiplicity of solutions for thep-Laplacian. Lecture Notes at the Second School on Nonlinear Functional Analysis and Applications to Differential Equations, ICTP, Trieste 1997

    1997

  48. [56]

    Pol´ aˇ cik, P

    P. Pol´ aˇ cik, P. Quittner and P. Souplet,Singularity and decay estimates in superlinear problems via Liouville- type theorems. I. Elliptic equations and systems, Duke Math. J.,139(2007), no. 3, 555-579

    2007

  49. [57]

    Roncoroni,Liouville-type results for the Lane-Emden equations, In Nonlocal and19nonlinear PDEs at the University of Bologna, volume 14(1) of Bruno Pini Math

    A. Roncoroni,Liouville-type results for the Lane-Emden equations, In Nonlocal and19nonlinear PDEs at the University of Bologna, volume 14(1) of Bruno Pini Math. Anal.20Semin., pages 77-94. Univ. Bologna, Alma Mater Stud., Bologna, 2023

    2023

  50. [58]

    Sciunzi,Classification of positiveD 1,p(RN)-solutions to the criticalp-Laplace equation inR N, Advances in Mathematics,291(2016), 12-23

    B. Sciunzi,Classification of positiveD 1,p(RN)-solutions to the criticalp-Laplace equation inR N, Advances in Mathematics,291(2016), 12-23

    2016

  51. [59]

    Serrin,Local behavior of solutions of quasi-linear equations, Acta Math.,111(1964), 247-302

    J. Serrin,Local behavior of solutions of quasi-linear equations, Acta Math.,111(1964), 247-302

    1964

  52. [60]

    Serrin,Singularities of solutions of nonlinear equations, Proc

    J. Serrin,Singularities of solutions of nonlinear equations, Proc. Sympos. Appl. Math., Vol. XVII pp. 68-88 Amer. Math. Soc., Providence, RI (1965)

    1965

  53. [61]

    Serrin and H

    J. Serrin and H. Zou,Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math.,189(2002), 79-142

    2002

  54. [62]

    Taylor,Crystalline variational problems, Bull

    J. Taylor,Crystalline variational problems, Bull. Amer. Math. Soc.,84(1978), 568-588

    1978

  55. [63]

    Tolksdorf,Regularity for more general class of quasilinear elliptic equations, J

    P. Tolksdorf,Regularity for more general class of quasilinear elliptic equations, J. Differential Equations,51 (1984), 126-150

    1984

  56. [64]

    Trudinger,On the Hanack type inequalities and their application to quasilinear elliptic equations, Comm

    N. Trudinger,On the Hanack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math.,20(2006), 159-177

    2006

  57. [65]

    V´ etois,A note on the classification of positive solutions to the criticalp-Laplace equation inR N, Adv

    J. V´ etois,A note on the classification of positive solutions to the criticalp-Laplace equation inR N, Adv. Nonlinear Stud.24(2024), 543-552

    2024

  58. [66]

    V´ etois,A priori estimates and application to the symmetry of solutions for criticalp-Laplace equations, J

    J. V´ etois,A priori estimates and application to the symmetry of solutions for criticalp-Laplace equations, J. Differential Equations,260(2016), 149-161

    2016

  59. [67]

    Wang and C

    G. Wang and C. Xia,A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal.,199(2011), 99-115

    2011

  60. [68]

    Wang and C

    G. Wang and C. Xia,Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations,252(2012), 1668-1700

    2012

  61. [69]

    Wei and X

    J. Wei and X. Xu,Classification of solutions of higher order conformally invariant equations, Math. Ann., 313(1999), no. 2, 207-228

    1999

  62. [70]

    Wulff,Zur Frage der geschwindigkeit des wachstums und der aufl¨ osung der kristallfl¨ aschen, Z

    G. Wulff,Zur Frage der geschwindigkeit des wachstums und der aufl¨ osung der kristallfl¨ aschen, Z. Kristallogr., 34(1901), 449-530

    1901

  63. [71]

    Xie and H

    R. Xie and H. Gong,A priori estimates and blow-up behavior for solutions of−Q N u=V e u in bounded domain inR N, Sci. China Math.,59(2016), 479-492

    2016

  64. [72]

    C. L. Zhou and C. Q. Zhou,Moser-Trudinger inequality involving the anisotropic Dirichlet normR Ω F N(∇u)dx 1 N onW 1,N 0 (Ω), J. Funct. Anal.,276(2019), 2901-2935

    2019

  65. [73]

    Zhou,Classification theorem for positive critical points of Sobolev trace inequality, arXiv: 2402.17602v3

    Y. Zhou,Classification theorem for positive critical points of Sobolev trace inequality, arXiv: 2402.17602v3. Key laboratory of Algebraic Lie Theory and Analysis of Ministry of Education, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P....

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