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January  2020, 19(1): 527-539. doi: 10.3934/cpaa.2020026

Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian

Phuong Le 1,2,

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

Received  February 2019 Revised  April 2019 Published  July 2019

Fund Project: This work was done while the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) in 2019. He wish to thank the institute for their hospitality and support.

Let
$ u \in L_{sp} \cap C^{1, 1}_{\rm loc}(\mathbb{R}^n\setminus\{0\}) $
 be a positive solution, which may blow up at zero, of the equation
$ (-\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\}, $
where
$ 0 < s < 1 $
,
$ 0 < \beta < n $
,
$ p>2 $
,
$ q\ge 1 $
 and
$ \alpha>0 $
. We prove that if
$ u $
 satisfies some suitable asymptotic properties, then
$ u $
 must be radially symmetric and monotone decreasing about the origin. In stead of using equivalent fractional systems, we exploit a direct method of moving planes for the weighted Choquard nonlinearity. This method allows us to cover the full range
$ 0 < \beta < n $
 in our results.
Keywords: Choquard equations, fractional p-Laplacian, symmetry of solutions, positive solutions, singular solutions.
Mathematics Subject Classification: 35R11, 35J92, 35J75, 35B06.
Citation: Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026
References:
[1]

G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, preprint, arXiv: 1810.11759. Google Scholar

[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. Google Scholar

[13]

E. P. Gross, Physics of Many-Particle Systems, Vol.1, Gordon Breach, New York, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P. Ma and J. Zhang, Symmetry and Nonexistence of Positive Solutions for Fractional Choquard Equations, preprint, arXiv: 1704.02190. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, Berlin, 1954. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, preprint, arXiv: 1810.11759. Google Scholar

[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. Google Scholar

[13]

E. P. Gross, Physics of Many-Particle Systems, Vol.1, Gordon Breach, New York, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P. Ma and J. Zhang, Symmetry and Nonexistence of Positive Solutions for Fractional Choquard Equations, preprint, arXiv: 1704.02190. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, Berlin, 1954. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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