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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

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.
Citation: 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
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. BelchiorH. BuenoO. 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. BjorlandL. 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. BjorlandL. 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. ChenC. 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. ChenC. 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. DaiY. 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. BelchiorH. BuenoO. 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. BjorlandL. 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. BjorlandL. 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. ChenC. 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. ChenC. 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. DaiY. 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.

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