Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian

Lihong Zhang; Wenwen Hou; Bashir Ahmad; Guotao Wang

doi:10.3934/dcdss.2020445

Previous Article
DCDS-S Home
This Issue
Next Article
On the observability of conformable linear time-invariant control systems

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 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.

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.

[1]	Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015
[2]	Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235
[3]	Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082
[4]	Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462
[5]	Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154
[6]	Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026
[7]	Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201
[8]	Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[9]	Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
[10]	Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068
[11]	CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure and Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004
[12]	Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319
[13]	Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[14]	Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447
[15]	Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
[16]	Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265
[17]	Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[18]	Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[19]	Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055
[20]	L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure and Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors