# American Institute of Mathematical Sciences

February  2021, 41(2): 489-505. doi: 10.3934/dcds.2020265

## Symmetry and nonexistence results for a fractional Choquard equation with weights

 1 Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam 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 3 Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

* Corresponding author: Phuong Le (lephuong@tdtu.edu.vn)

Received  August 2019 Published  February 2021 Early access  July 2020

Let
 $u$
be a nonnegative solution to the equation
 $(-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * |x|^a u^p \right) |x|^a u^{p-1} \quad\text{ in } \mathbb{R}^n \setminus \{0\},$
where
 $n \ge 2$
,
 $0 < \alpha < 2$
,
 $0 < \beta < n$
and
 $a>\max\{ -\alpha, -\frac{\alpha+\beta}{2} \}$
. By exploiting the method of scaling spheres and moving planes in integral forms, we show that
 $u$
must be zero if
 $1\le p<\frac{n+\beta+2a}{n-\alpha}$
 $a<0$
and
 $\frac{n+\beta+2a}{n-\alpha} \le p \le \frac{n+\beta+a}{n-\alpha}$
.
Citation: 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
##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, vol. 116 of Cambridge Studies in Advanced Mathematics, 2nd edition, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511809781. [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] J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996. [4] J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N. [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] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903. [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] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, vol. 1871 of Lecture Notes in Math., Springer, Berlin, 2006, 1–43. doi: 10.1007/11545989_1. [10] 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. [11] W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres, Preprint, arXiv: 1810.02752. [12] 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. [13] L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, Preprint, arXiv: 1810.11759. [14] T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. [15] 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. [16] P. Le, Symmetry of singular solutions for a weighted Choquard equation involving the fractional p-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 527-539.  doi: 10.3934/cpaa.2020026. [17] Y. Lei, Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., 38 (2018), 5351-5377.  doi: 10.3934/dcds.2018236. [18] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293. [19] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.  doi: 10.2307/2007032. [20] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845. [21] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [22] 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. [23] P. Ma, X. Shang and J. Zhang, Symmetry and nonexistence of positive solutions for fractional Choquard equations, Pacific J. Math., 304 (2020), 143-167.  doi: 10.2140/pjm.2020.304.143. [24] 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. [25] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, vol. 162 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2016, With a foreword by Jean Mawhin. doi: 10.1017/CBO9781316282397. [26] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733–2742, Topology of the Universe Conference (Cleveland, OH, 1997). doi: 10.1088/0264-9381/15/9/019. [27] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1. [28] G. I. Nazin, Limit distribution functions of systems with many-particle interactions in classical statistical physics, Teoret. Mat. Fiz., 25 (1975), 132-140. [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [30] 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. [31] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

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##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, vol. 116 of Cambridge Studies in Advanced Mathematics, 2nd edition, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511809781. [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] J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996. [4] J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N. [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] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903. [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] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, vol. 1871 of Lecture Notes in Math., Springer, Berlin, 2006, 1–43. doi: 10.1007/11545989_1. [10] 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. [11] W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres, Preprint, arXiv: 1810.02752. [12] 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. [13] L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, Preprint, arXiv: 1810.11759. [14] T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. [15] 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. [16] P. Le, Symmetry of singular solutions for a weighted Choquard equation involving the fractional p-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 527-539.  doi: 10.3934/cpaa.2020026. [17] Y. Lei, Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., 38 (2018), 5351-5377.  doi: 10.3934/dcds.2018236. [18] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293. [19] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.  doi: 10.2307/2007032. [20] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845. [21] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [22] 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. [23] P. Ma, X. Shang and J. Zhang, Symmetry and nonexistence of positive solutions for fractional Choquard equations, Pacific J. Math., 304 (2020), 143-167.  doi: 10.2140/pjm.2020.304.143. [24] 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. [25] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, vol. 162 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2016, With a foreword by Jean Mawhin. doi: 10.1017/CBO9781316282397. [26] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733–2742, Topology of the Universe Conference (Cleveland, OH, 1997). doi: 10.1088/0264-9381/15/9/019. [27] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1. [28] G. I. Nazin, Limit distribution functions of systems with many-particle interactions in classical statistical physics, Teoret. Mat. Fiz., 25 (1975), 132-140. [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [30] 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. [31] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.
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