# 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  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
 $1\le p<\frac{n+\beta+2a}{n-\alpha}$
and must be radially symmetric about the origin if
 $a<0$
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 & Continuous Dynamical Systems - A, 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.  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] J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.   Google Scholar [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.  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] 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.  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] 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [13] L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, Preprint, arXiv: 1810.11759. Google Scholar [14] T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] G. I. Nazin, Limit distribution functions of systems with many-particle interactions in classical statistical physics, Teoret. Mat. Fiz., 25 (1975), 132-140.   Google Scholar [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.   Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  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] J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.   Google Scholar [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.  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] 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.  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] 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [13] L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choquard equations, Preprint, arXiv: 1810.11759. Google Scholar [14] T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] G. I. Nazin, Limit distribution functions of systems with many-particle interactions in classical statistical physics, Teoret. Mat. Fiz., 25 (1975), 132-140.   Google Scholar [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.   Google Scholar [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.  Google Scholar [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.  Google Scholar
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