On some Liouville theorems for $ p $-Laplace type operators

Michel Chipot; Daniel Hauer; Michel Chipot; Daniel Hauer

doi:10.3934/cam.2025039

Communications in Analysis and Mechanics

2025, Volume 17, Issue 4: 955-970. doi: 10.3934/cam.2025039

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On some Liouville theorems for $ p $-Laplace type operators

Michel Chipot ^{1
,
,},
Daniel Hauer ^{2,3
,
,}

1.
Institute of Mathematics, University of Zürich, Winterthurerstr.190, 8057 Zürich, Switzerland
2.
Brandenburg University of Technology Cottbus-Senftenberg, Faculty 1 - Section Analysis, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany
3.
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, 2006, Australia

Received: 13 November 2024 Revised: 11 July 2025 Accepted: 02 September 2025 Published: 03 December 2025
35A01, 35B53, 35D30, 35F25

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.
- $ p $-Laplace operator,
- Liouville theorem,
- Schrödinger equation,
- nonlinear operators,
- anisotropic Laplace operator,
- double phase problem
Citation: Michel Chipot, Daniel Hauer. On some Liouville theorems for $ p $-Laplace type operators[J]. Communications in Analysis and Mechanics, 2025, 17(4): 955-970. doi: 10.3934/cam.2025039

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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.

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 doi: 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 doi: 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 doi: 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 doi: 10.1007/s10958-011-0260-7
[10]	B. Brandolini, F. C. Cîrstea, Anisotropic elliptic equations with gradient-dependent lower order terms and $L^1$ data, Math. Eng., 5 (2023), 1–33. https://doi.org/10.3934/mine.2023073 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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.

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