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: |
[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.![]() ![]() ![]() |