On some Liouville theorems for p-Laplace type operators

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Abstract

The aim of this note is to examine Liouville-type theorems for p-Laplacian-type operators. Guided by the Laplacian case, analogous results are established for the p-Laplacian and sums of operators of this type.

Keywords

p-Laplace operator / Liouville theorem / Schrödinger equation / nonlinear operators / anisotropic Laplace operator / double phase problem

CHANGE HISTORY

A Correction to this paper has been published:
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The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
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References

[1] L. C. Evans, Partial Differential Equations , Volume 19 of Graduate Studies in Mathematics, American Math. Society, Providence, 1998.

[2] M. H. Protter, H. F. Weinberger, Maximum Principles in Differential Equations , Prentice-Hall, Englewood Cliffs, NJ, 1967.

[3] H. Brezis, M. Chipot, Y. Xie, Some remarks on Liouville type theorems, In World Scientific Edt., editor, Proceedings of the international conference in nonlinear analysis, 43-65, Hsinchu, Taiwan 2006, 2008.

[4] A. Grigor’yan, Bounded solutions of the Schrödinger equation on non-compact Riemannian manifolds, J. Sov. Math., 51 (1990), 2340-2349. https://doi.org/10.1007/BF01094993

[5] A. Grigor’yan, W. Hansen, A Liouville property for Schrödinger operators, Math. Ann., 312 (1998), 659-716. https://doi.org/10.1007/s002080050241

[6] R. G. Pinsky, A probabilistic approach to a Liouville-type problem for Schrödinger operators, Preprint, 2006.

[7] P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), 48. https://doi.org/10.1007/s00526-018-1332-z

[8] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions , J. Differential Equations, 90 (1991), 1-30.

[9] V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci., 173 (2011), 463-570. https://doi.org/10.1007/s10958-011-0260-7

[10] B. Brandolini, F. C. Cȋrstea, Anisotropic elliptic equations with gradient-dependent lower order terms and L¹ data , Math. Eng., 5 (2023), 1-33. https://doi.org/10.3934/mine.2023073

[11] E. N. Dancer, D. Daners, D. Hauer, A Liouville theorem for p-harmonic functions on exterior domains, Positivity, 19 (2015), 577-586. https://doi.org/10.1007/s11117-014-0316-2

[12] E. Mitidieri, S. I. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129-162. https://doi.org/10.1007/s00032-004-0032-7

[13] I. Birindelli, F. Demengel, Some Liouville theorems for the p-Laplacian , Proceedings of the 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, Electron. J. Differ. Equ. Conf., 8 (2002), 35-46.

[14] M. Chipot, Elliptic Equations: An Introductory Course, Second edition , Birkhäuser, 2024. https://doi.org/10.1007/978-3-031-54123-0

[15] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations , World Scientific, Taipei, 2006.

[16] A. Farina, C. Mercuri, M. Willem, A Liouville theorem for the p-Laplacian and related questions , Calc. Var. Partial Differential Equations, 58 (2019), 13. https://doi.org/10.1007/s00526-019-1596-y

[17] M. Meier, Liouville theorem for nonlinear elliptic equations and systems, Manuscripta Math., 29 (1979), 207-228. https://doi.org/10.1007/BF01303628

[18] P. Pucci, J. Serrin, The Maximum Principle , volume 73 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 2007.

[19] P. Quittner, P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States , Birkhäuser, 2007.

[20] Y. Pinchover, A. Tertikas, K. Tintarev, A Liouville-type theorem for the p-laplacian with potential terms, Ann. I. H. Poincaré, 25 (2008), 357-368. https://doi.org/10.1016/j.anihpc.2006.12.004

[21] J. Vétois, A priori estimates for solutions of anisotropic elliptic equations, Nonlinear Anal., 71 (2009), 3881-3905. https://doi.org/10.1016/j.na.2009.02.076

[22] J. Vétois, Strong maximum principles for anisotropic elliptic and parabolic equations, Adv. Nonlinear Stud., 12 (2012), 101-114.

[23] M. Chipot, Elliptic Equations: An Introductory Course , Birkhäuser, 2009. https://doi.org/10.1007/978-3-7643-9982-5

[24] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and their Applications , volume 31 of Classic Appl. Math. SIAM, Philadelphia, 2000.