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February  2020, 19(2): 771-783. doi: 10.3934/cpaa.2020036

## Liouville theorems for an integral equation of Choquard type

 1 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Received  November 2018 Revised  July 2019 Published  October 2019

We establish sharp Liouville theorems for the integral equation
 $u(x) = \int_{\mathbb{R}^n} \frac{u^{p-1}(y)}{|x-y|^{n-\alpha}} \int_{\mathbb{R}^n} \frac{u^p(z)}{|y-z|^{n-\beta}} dz dy, \quad x\in\mathbb{R}^n,$
where
 $0<\alpha, \beta and $ p>1 $. Our results hold true for positive solutions under appropriate assumptions on $ p $and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive $ H^{\frac{\alpha}{2}}(\mathbb{R}^n) $solutions of the higher fractional order Choquard type equation $ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * u^p\right) u^{p-1} \quad\text{ in } \mathbb{R}^n. $Citation: Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure and Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036 ##### References:  [1] D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [2] 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. [3] 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. [4] J. Bertoin, Lévy Processes, Cambridge University Press, 1996. [5] J. P. Bouchard 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. [6] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [7] D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994. doi: 10.1017/prm.2018.67. [8] G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3. [9] 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. [10] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [11] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci., 29B (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [12] P. Constantin, Euler Equations, Navier-Stokes Equations and Turbulence, Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of lecture Notes in Math. 1–43, Springer, Berlin, 2006. doi: 10.1007/11545989_1. [13] W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403. doi: 10.3934/dcds.2018117. [14] P. Le, Symmetry and classification of solutions to an integral equation of Choquard type, submitted for publication. [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] 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. [17] Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905. doi: 10.1007/s00209-012-1036-6. [18] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374. doi: 10.2307/2007032. [19] E. Lieb, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977) 185–194. [20] 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. [21] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3. [22] I. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019. [23] 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. [24] V. Moroz and J. V. Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains}, J. Differential Equations, 254 (2013), 3089-3145. doi: 10.1016/j.jde.2012.12.019. [25] S. Pekar, Untersuchungen über die Elekronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. [26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970. [27] V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889. doi: 10.1016/j.cnsns.2006.03.005. [28] D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89. doi: 10.1016/j.aml.2014.12.007. [29] 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] D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [2] 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. [3] 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. [4] J. Bertoin, Lévy Processes, Cambridge University Press, 1996. [5] J. P. Bouchard 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. [6] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [7] D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994. doi: 10.1017/prm.2018.67. [8] G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3. [9] 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. [10] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [11] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci., 29B (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [12] P. Constantin, Euler Equations, Navier-Stokes Equations and Turbulence, Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of lecture Notes in Math. 1–43, Springer, Berlin, 2006. doi: 10.1007/11545989_1. [13] W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403. doi: 10.3934/dcds.2018117. [14] P. Le, Symmetry and classification of solutions to an integral equation of Choquard type, submitted for publication. [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] 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. [17] Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905. doi: 10.1007/s00209-012-1036-6. [18] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374. doi: 10.2307/2007032. [19] E. Lieb, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977) 185–194. [20] 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. [21] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3. [22] I. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019. [23] 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. [24] V. Moroz and J. V. Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains}, J. Differential Equations, 254 (2013), 3089-3145. doi: 10.1016/j.jde.2012.12.019. [25] S. Pekar, Untersuchungen über die Elekronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. [26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970. [27] V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889. doi: 10.1016/j.cnsns.2006.03.005. [28] D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89. doi: 10.1016/j.aml.2014.12.007. [29] 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.  [1] 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 [2] César E. Torres Ledesma. Existence of positive solutions for a class of fractional Choquard equation in exterior domain. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3301-3328. doi: 10.3934/dcds.2022016 [3] Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2237-2256. doi: 10.3934/cpaa.2021065 [4] Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional$ p $-Laplacian. 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