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On the observability of conformable linear time-invariant control systems
Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian
a. | School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China |
b. | Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia |
In this paper, we investigate radial symmetry and monotonicity of positive solutions to a logarithmic Choquard equation involving a generalized nonlinear tempered fractional $ p $-Laplacian operator by applying the direct method of moving planes. We first introduce a new kind of tempered fractional $ p $-Laplacian $ (-\Delta-\lambda_{f})_{p}^{s} $ based on tempered fractional Laplacian $ (\Delta+\lambda)^{\beta/2} $, which was originally defined in [
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J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. |
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W. Chen and C. Li,
Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.
doi: 10.1016/j.aim.2018.07.016. |
[3] |
W. Deng, B. Li, W. Tian and P. Zhang,
Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul., 16 (2018), 125-149.
doi: 10.1137/17M1116222. |
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P. d'Avenia, G. Siciliano and M. Squassina,
On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.
doi: 10.1142/S0218202515500384. |
[5] |
S. Duo and Y. Zhang,
Numerical approximations for the tempered fractional Laplacian: Error analysis and applications, J. Sci. Comput., 81 (2019), 569-593.
doi: 10.1007/s10915-019-01029-7. |
[6] |
D. Kumar, J. Singh and D. Baleanu,
A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Thermal Science, 22 (2018), 2791-2802.
doi: 10.2298/TSCI170129096K. |
[7] |
D. Kumar, J. Singh and D. Baleanu,
On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods Appl. Sci., 43 (2020), 443-457.
doi: 10.1002/mma.5903. |
[8] |
C. Li, W. Deng and L. Zhao,
Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1989-2015.
doi: 10.3934/dcdsb.2019026. |
[9] |
V. Moroz and J. V. Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[10] |
P. Mohammed, M. Sarikaya and D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, 12 (2020), 595.
doi: 10.3390/sym12040595. |
[11] |
L. Ma and Z. Zhang,
Symmetry of positive solutions for Choquard equations with fractional $p$-Laplacian, Nonlinear Anal., 182 (2019), 248-262.
doi: 10.1016/j.na.2018.12.015. |
[12] |
H. M. Srivastava, V. P. Dubey, R. Kumar, J. Singh, D. Kumar and D. Baleanu, An efficient computational approach for a fractional-order biological population model with carrying capacity, Chaos Solitons Fractals, 138 (2020), 109880, 13 pp.
doi: 10.1016/j.chaos.2020.109880. |
[13] |
J. Sun, D. Nie and W. Deng, Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian, preprint, 2018, arXiv: 1802.02349. |
[14] |
B. Shiri, G. Wu and D. Baleanu,
Collocation methods for terminal value problems of tempered fractional differential equations, Appl. Numer. Math., 156 (2020), 385-395.
doi: 10.1016/j.apnum.2020.05.007. |
[15] |
G. Wang, X. Ren, Z. Bai and W. Hou,
Radial symmetry of standing waves for nonlinear fractional Hardy-Schrödinger equation, Appl. Math. Lett., 96 (2019), 131-137.
doi: 10.1016/j.aml.2019.04.024. |
[16] |
G. Wang and X. Ren, Radial symmetry of standing waves for nonlinear fractional Laplacian Hardy-Schrödinger systems, Appl. Math. Lett., 110 (2020), 106560, 8 pp.
doi: 10.1016/j.aml.2020.106560. |
[17] |
L. Zhang, B. Ahmad, G. Wang and X. Ren, Radial symmetry of solution for fractional $p-$Laplacian system, Nonlinear Anal., 196 (2020), 111801, 16 pp.
doi: 10.1016/j.na.2020.111801. |
[18] |
Z. Zhang, W. Deng and H. Fan, Finite difference schemes for the tempered fractional Laplacian, Numer. Math. Theory Methods Appl. 12 (2019), 492-–516.
doi: 10.4208/nmtma.OA-2017-0141. |
[19] |
Z. Zhang, W. Deng and GE. Karniadakis,
A Riesz basis Galerkin method for the tempered fractional Laplacian, SIAM J. Numer. Anal., 56 (2018), 3010-3039.
doi: 10.1137/17M1151791. |
[20] |
L. Zhang and W. Hou, Standing waves of nonlinear fractional p-Laplacian Schrödinger equation involving logarithmic nonlinearity, Appl. Math. Lett., 102 (2020), 106149, 6 pp.
doi: 10.1016/j.aml.2019.106149. |
show all references
References:
[1] |
J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. |
[2] |
W. Chen and C. Li,
Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.
doi: 10.1016/j.aim.2018.07.016. |
[3] |
W. Deng, B. Li, W. Tian and P. Zhang,
Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul., 16 (2018), 125-149.
doi: 10.1137/17M1116222. |
[4] |
P. d'Avenia, G. Siciliano and M. Squassina,
On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.
doi: 10.1142/S0218202515500384. |
[5] |
S. Duo and Y. Zhang,
Numerical approximations for the tempered fractional Laplacian: Error analysis and applications, J. Sci. Comput., 81 (2019), 569-593.
doi: 10.1007/s10915-019-01029-7. |
[6] |
D. Kumar, J. Singh and D. Baleanu,
A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Thermal Science, 22 (2018), 2791-2802.
doi: 10.2298/TSCI170129096K. |
[7] |
D. Kumar, J. Singh and D. Baleanu,
On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods Appl. Sci., 43 (2020), 443-457.
doi: 10.1002/mma.5903. |
[8] |
C. Li, W. Deng and L. Zhao,
Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1989-2015.
doi: 10.3934/dcdsb.2019026. |
[9] |
V. Moroz and J. V. Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[10] |
P. Mohammed, M. Sarikaya and D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, 12 (2020), 595.
doi: 10.3390/sym12040595. |
[11] |
L. Ma and Z. Zhang,
Symmetry of positive solutions for Choquard equations with fractional $p$-Laplacian, Nonlinear Anal., 182 (2019), 248-262.
doi: 10.1016/j.na.2018.12.015. |
[12] |
H. M. Srivastava, V. P. Dubey, R. Kumar, J. Singh, D. Kumar and D. Baleanu, An efficient computational approach for a fractional-order biological population model with carrying capacity, Chaos Solitons Fractals, 138 (2020), 109880, 13 pp.
doi: 10.1016/j.chaos.2020.109880. |
[13] |
J. Sun, D. Nie and W. Deng, Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian, preprint, 2018, arXiv: 1802.02349. |
[14] |
B. Shiri, G. Wu and D. Baleanu,
Collocation methods for terminal value problems of tempered fractional differential equations, Appl. Numer. Math., 156 (2020), 385-395.
doi: 10.1016/j.apnum.2020.05.007. |
[15] |
G. Wang, X. Ren, Z. Bai and W. Hou,
Radial symmetry of standing waves for nonlinear fractional Hardy-Schrödinger equation, Appl. Math. Lett., 96 (2019), 131-137.
doi: 10.1016/j.aml.2019.04.024. |
[16] |
G. Wang and X. Ren, Radial symmetry of standing waves for nonlinear fractional Laplacian Hardy-Schrödinger systems, Appl. Math. Lett., 110 (2020), 106560, 8 pp.
doi: 10.1016/j.aml.2020.106560. |
[17] |
L. Zhang, B. Ahmad, G. Wang and X. Ren, Radial symmetry of solution for fractional $p-$Laplacian system, Nonlinear Anal., 196 (2020), 111801, 16 pp.
doi: 10.1016/j.na.2020.111801. |
[18] |
Z. Zhang, W. Deng and H. Fan, Finite difference schemes for the tempered fractional Laplacian, Numer. Math. Theory Methods Appl. 12 (2019), 492-–516.
doi: 10.4208/nmtma.OA-2017-0141. |
[19] |
Z. Zhang, W. Deng and GE. Karniadakis,
A Riesz basis Galerkin method for the tempered fractional Laplacian, SIAM J. Numer. Anal., 56 (2018), 3010-3039.
doi: 10.1137/17M1151791. |
[20] |
L. Zhang and W. Hou, Standing waves of nonlinear fractional p-Laplacian Schrödinger equation involving logarithmic nonlinearity, Appl. Math. Lett., 102 (2020), 106149, 6 pp.
doi: 10.1016/j.aml.2019.106149. |
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