# American Institute of Mathematical Sciences

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

## 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

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.

Citation: Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete and 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. [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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##### 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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