-
Previous Article
Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method
- CPAA Home
- This Issue
-
Next Article
Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator
Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam |
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam |
$ u \in L_{sp} \cap C^{1, 1}_{\rm loc}(\mathbb{R}^n\setminus\{0\}) $ |
$ (-\Delta)^s_p u = \left(\frac{1}{|x|^{n-\beta }} * \frac{u^q}{|x|^\alpha}\right) \frac{u^{q-1 }}{|x|^\alpha} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $ |
$ 0 < s < 1 $ |
$ 0 < \beta < n $ |
$ p>2 $ |
$ q\ge 1 $ |
$ \alpha>0 $ |
$ u $ |
$ u $ |
$ 0 < \beta < n $ |
References:
[1] |
G. Siciliano and M. Squassina,
On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.
doi: 10.1142/S0218202515500384. |
[2] |
P. Belchior, H. Bueno, O. H. Miyagaki and G. A. Pereira,
Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Anal., 164 (2017), 38-53.
doi: 10.1016/j.na.2017.08.005. |
[3] |
C. Bjorland, L. Caffarelli and A. Figalli,
Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.
doi: 10.1016/j.aim.2012.03.032. |
[4] |
C. Bjorland, L. Caffarelli and A. Figalli,
Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.
doi: 10.1002/cpa.21379. |
[5] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[6] |
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. |
[7] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[8] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[9] |
W. Dai, Y. Fang and G. Qin,
Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes, J. Differential Equations, 265 (2018), 2044-2063.
doi: 10.1016/j.jde.2018.04.026. |
[10] |
J. Dou and H. Zhou,
Liouville theorem for fractional Hénon equation and system on $\mathbb{R}^n$, Comm. Pure Appl. Anal., 14 (2015), 1915-1927.
doi: 10.3934/cpaa.2015.14.1915. |
[11] |
L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, preprint, arXiv: 1810.11759. |
[12] |
A. T. Duong and P. Le, Symmetry and nonexistence results for a fractional Hénon-Hardy system on a half-space, Rocky Mountain J. Math., (2019), to appear. Available from: https://projecteuclid.org/euclid.rmjm/1552186836. |
[13] |
E. P. Gross, Physics of Many-Particle Systems, Vol.1, Gordon Breach, New York, 1966. |
[14] |
P. Le, Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., 185 (2019), 123-141.
doi: 10.1016/j.na.2019.03.006. |
[15] |
E. H. Lieb and B. Simon,
The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.
doi: 10.1007/BF01609845. |
[16] |
B. Liu and L. Ma,
Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135.
doi: 10.1016/j.na.2016.08.022. |
[17] |
S. Liu,
Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.
doi: 10.1016/j.na.2009.01.014. |
[18] |
P. Ma and J. Zhang, Symmetry and Nonexistence of Positive Solutions for Fractional Choquard Equations, preprint, arXiv: 1704.02190. |
[19] |
P. Ma and J. Zhang,
Existence and multiplicity of solutions for fractional Choquard equations, Nonlinear Anal., 164 (2017), 100-117.
doi: 10.1016/j.na.2017.07.011. |
[20] |
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. |
[21] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[22] |
V. Moroz and J. V. Schaftingen,
A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.
doi: 10.1007/s11784-016-0373-1. |
[23] |
S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, Berlin, 1954. |
[24] |
L. Wu and P. Niu,
Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations, Discrete Contin. Dyn. Syst., 39 (2018), 1573-1583.
doi: 10.3934/dcds.2019069. |
[25] |
D. Xu and Y. Lei,
Classification of positive solutions for a static Schrodinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.
doi: 10.1016/j.aml.2014.12.007. |
[26] |
W. Zhang and X. Wu,
Nodal solutions for a fractional Choquard equation, J. Math. Anal. Appl., 464 (2018), 1167-1183.
doi: 10.1016/j.jmaa.2018.04.048. |
show all references
References:
[1] |
G. Siciliano and M. Squassina,
On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.
doi: 10.1142/S0218202515500384. |
[2] |
P. Belchior, H. Bueno, O. H. Miyagaki and G. A. Pereira,
Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Anal., 164 (2017), 38-53.
doi: 10.1016/j.na.2017.08.005. |
[3] |
C. Bjorland, L. Caffarelli and A. Figalli,
Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.
doi: 10.1016/j.aim.2012.03.032. |
[4] |
C. Bjorland, L. Caffarelli and A. Figalli,
Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.
doi: 10.1002/cpa.21379. |
[5] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[6] |
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. |
[7] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[8] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[9] |
W. Dai, Y. Fang and G. Qin,
Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes, J. Differential Equations, 265 (2018), 2044-2063.
doi: 10.1016/j.jde.2018.04.026. |
[10] |
J. Dou and H. Zhou,
Liouville theorem for fractional Hénon equation and system on $\mathbb{R}^n$, Comm. Pure Appl. Anal., 14 (2015), 1915-1927.
doi: 10.3934/cpaa.2015.14.1915. |
[11] |
L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, preprint, arXiv: 1810.11759. |
[12] |
A. T. Duong and P. Le, Symmetry and nonexistence results for a fractional Hénon-Hardy system on a half-space, Rocky Mountain J. Math., (2019), to appear. Available from: https://projecteuclid.org/euclid.rmjm/1552186836. |
[13] |
E. P. Gross, Physics of Many-Particle Systems, Vol.1, Gordon Breach, New York, 1966. |
[14] |
P. Le, Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., 185 (2019), 123-141.
doi: 10.1016/j.na.2019.03.006. |
[15] |
E. H. Lieb and B. Simon,
The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.
doi: 10.1007/BF01609845. |
[16] |
B. Liu and L. Ma,
Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135.
doi: 10.1016/j.na.2016.08.022. |
[17] |
S. Liu,
Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.
doi: 10.1016/j.na.2009.01.014. |
[18] |
P. Ma and J. Zhang, Symmetry and Nonexistence of Positive Solutions for Fractional Choquard Equations, preprint, arXiv: 1704.02190. |
[19] |
P. Ma and J. Zhang,
Existence and multiplicity of solutions for fractional Choquard equations, Nonlinear Anal., 164 (2017), 100-117.
doi: 10.1016/j.na.2017.07.011. |
[20] |
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. |
[21] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[22] |
V. Moroz and J. V. Schaftingen,
A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.
doi: 10.1007/s11784-016-0373-1. |
[23] |
S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, Berlin, 1954. |
[24] |
L. Wu and P. Niu,
Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations, Discrete Contin. Dyn. Syst., 39 (2018), 1573-1583.
doi: 10.3934/dcds.2019069. |
[25] |
D. Xu and Y. Lei,
Classification of positive solutions for a static Schrodinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.
doi: 10.1016/j.aml.2014.12.007. |
[26] |
W. Zhang and X. Wu,
Nodal solutions for a fractional Choquard equation, J. Math. Anal. Appl., 464 (2018), 1167-1183.
doi: 10.1016/j.jmaa.2018.04.048. |
[1] |
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 |
[2] |
Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1121-1147. doi: 10.3934/dcdsb.2021083 |
[3] |
Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922 |
[4] |
Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063 |
[5] |
Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure and Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019 |
[6] |
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 |
[7] |
Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593 |
[8] |
Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060 |
[9] |
Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079 |
[10] |
Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002 |
[11] |
Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 143-158. doi: 10.3934/dcds.2007.17.143 |
[12] |
Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055 |
[13] |
Yunyun Hu. Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3723-3734. doi: 10.3934/cpaa.2020164 |
[14] |
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 |
[15] |
Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623 |
[16] |
Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1031-1051. doi: 10.3934/cpaa.2009.8.1031 |
[17] |
Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033 |
[18] |
Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595 |
[19] |
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 |
[20] |
Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]