Section
|
Mathematics
|
Title
|
Liouville type theorems for solutions of semilinear equations on non-compact Riemannian manifolds
|
Author(-s)
|
Losev A.G.a,
Filatov V.V.a
|
Affiliations
|
Volgograd State Universitya
|
Abstract
|
It is proved that the Liouville function associated with the semilinear equation $\Delta u -g(x,u)=0$ is identical to zero if and only if there is only a trivial bounded solution of the semilinear equation on non-compact Riemannian manifolds. This result generalizes the corresponding result of S.A. Korolkov for the case of the stationary Schrödinger equation $ \Delta u-q (x) u = 0$. The concept of the capacity of a compact set associated with the stationary Schrödinger equation is also introduced and it is proved that if the capacity of any compact set is equal to zero, then the Liouville function is identically zero.
|
Keywords
|
Liouville type theorem, semilinear elliptic equations, Riemannian manifolds, massive sets, Liouville function
|
UDC
|
517.956.2
|
MSC
|
58J05
|
DOI
|
10.35634/vm210407
|
Received
|
6 July 2021
|
Language
|
English
|
Citation
|
Losev A.G., Filatov V.V. Liouville type theorems for solutions of semilinear equations on non-compact Riemannian manifolds, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 4, pp. 629-639.
|
References
|
- Caristi G., Mitidieri E., Pohozaev S.I. Some Liouville theorems for quasilinear elliptic inequalities, Doklady Mathematics, 2009, vol. 79, no. 1, pp. 118-124. https://doi.org/10.1134/S1064562409010360
- Cheng S.Y., Yau S.T. Differential equations on Riemannian manifolds and their geometric applications, Communications on Pure and Applied Mathematics, 1975, vol. 28, no. 3, pp. 333-354. https://doi.org/10.1002/cpa.3160280303
- Grigor'yan A. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bulletin of the American Mathematical Society, 1999, vol. 36, no. 2, pp. 135-249. https://doi.org/10.1090/S0273-0979-99-00776-4
- Grigor'yan A.A. On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds, Mathematics of the USSR-Sbornik, 1987, vol. 56, no. 2, pp. 349-358. https://doi.org/10.1070/SM1987v056n02ABEH003040
- Grigor'yan A., Hansen W. A Liouville property for Schrödinger operators, Mathematische Annalen, 1998, vol. 312, no. 4, pp. 659-716. https://doi.org/10.1007/s002080050241
- Grigor'yan A.A., Losev A.G. Dimension of spaces of solutions of the Schrödinger equation on noncompact Riemannian manifolds, Mathematical Physics and Computer Simulation, 2017, vol. 20, no. 3, pp. 34-42 (in Russian). https://doi.org/10.15688/mpcm.jvolsu.2017.3.3
- Grigor'yan A., Sun Y. On positive solutions of semi-linear elliptic inequalities on Riemannian manifolds, Calculus of Variations and Partial Differential Equations, 2019, vol. 58, no. 6, 207. https://doi.org/10.1007/s00526-019-1645-6
- Grigor'yan A., Verbitsky I. Pointwise estimates of solutions to semilinear elliptic equations and inequalities, Journal d'Analyse Mathématique, 2019, vol. 137, no 2, pp. 559-601. https://doi.org/10.1007/s11854-019-0004-z
- Keselman V.M. The concept and criteria of the capacitive type of the non-compact Riemannian manifold based on the generalized capacity, Mathematical Physics and Computer Simulation, 2019, vol. 22, no. 2, pp. 21-32 (in Russian). https://doi.org/10.15688/mpcm.jvolsu.2019.2.2
- Korol'kov S.A. Harmonic functions on Riemannian manifolds with ends, Abstract of Cand. Sci. (Phys.-Math.) Dissertation, Kazan, 2009, 18 p. (In Russian). https://www.elibrary.ru/item.asp?id=15942555
- Korolkov S.A., Losev A.G. Generalized harmonic functions of Riemannian manifolds with ends, Mathematische Zeitschrift, 2012, vol. 272, no. 1-2, pp. 459-472. https://doi.org/10.1007/s00209-011-0943-2
- Li P., Tam L.-F. Harmonic functions and the structure of complete manifolds, Journal of Differential Geometry, 1992, vol. 35, no. 2, pp. 359-383. https://doi.org/10.4310/jdg/1214448079
- Losev A.G., Filatov V.V. On capacitary characteristics of noncompact Riemannian manifolds, Russian Mathematics, 2021, vol. 65, no. 3, pp. 61-67. https://doi.org/10.3103/S1066369X21030063
- Mazepa E.A. On the solvability of boundary value problems for quasilinear elliptic equations on noncompact Riemannian manifolds, Sibirskie Élektronnye Matematicheskie Izvestiya, 2016, vol. 13, pp. 1026-1034 (in Russian). https://doi.org/10.17377/semi.2016.13.081
- Mazepa E.A. The Liouville property and boundary value problems for semilinear elliptic equations on noncompact Riemannian manifolds, Siberian Mathematical Journal, 2012, vol. 53, no. 1, pp. 134-145. https://doi.org/10.1134/S0037446612010119
- Miklyukov V.M. Some criteria for parabolicity and hyperbolicity of the boundary sets of surfaces, Izvestiya: Mathematics, 1996, vol. 60, no. 4, pp. 763-809. https://doi.org/10.1070/IM1996v060n04ABEH000080
- Murata M., Tsuchida T. Uniqueness of $L$$1$ harmonic functions on rotationally symmetric Riemannian manifolds, Kodai Mathematical Journal, 2014, vol. 37, no. 1, pp. 1-15. https://doi.org/10.2996/kmj/1396008245
- Sun Y. Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds, Journal of Mathematical Analysis and Applications, 2014, vol. 419, no. 1, pp. 643-661. https://doi.org/10.1016/j.jmaa.2014.05.011
- Yau S.-T. Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana University Mathematics Journal, 1976, vol. 25, no. 7, pp. 659-670. http://www.jstor.org/stable/24891285
|
Full text
|
|