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October  2021, 14(10): 3851-3863. doi: 10.3934/dcdss.2020445

Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian

Lihong Zhang a, , Wenwen Hou a, , Bashir Ahmad b,, and  Guotao Wang a,b,

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

* Corresponding author: Bashir Ahmad

Received  May 2020 Revised  July 2020 Published  October 2021 Early access  November 2020

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 [3] by Deng et.al [Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul., 16(1)(2018), 125-149]. Then we discuss the decay of solutions at infinity and narrow region principle, which play a key role in obtaining the main result by the process of moving planes.

Keywords: Generalized tempered fractional $ p $-Laplacian, direct method of moving planes, Choquard equations, logarithmic nonlinearity, radial symmetry and monotonicity.
Mathematics Subject Classification: 35A01, 35A30, 35B09.
Citation: Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445
References:
[1]

J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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