# American Institute of Mathematical Sciences

September  2016, 15(5): 1825-1840. doi: 10.3934/cpaa.2016017

## Liouville type theorems for singular integral equations and integral systems

 1 Institute for Advanced Study, Shenzhen University, Shenzhen Guangdong, 518060

Received  October 2015 Revised  February 2016 Published  July 2016

In this paper, we establish some Liouville type theorems for positive solutions of some integral equations and integral systems in $R^N$. The main technique we use is the method of moving planes in an integral form.
Citation: Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017
##### References:
 [1] H. Berestycki, L. A. Caffarelli and L. Nireberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Norm. Sup. Pisa. C1. Sci., 25 (1997), 69-94. [2] G. Bianchi, Nonexistence of positive solutions to semilinear elliptic equations on $R^N$ or $R^N_+$ through the method of moving planes, Comm. P.D.E., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315. [3] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure App. Math., XLII (1989), 271-297. doi: 10.1002/cpa.3160420304. [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [5] W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [6] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [7] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series, vol. 4, 2010. [8] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [9] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., LIX (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. P.D.E., 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [11] L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86. [12] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Advances in Mathematics, 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [13] D. G. De Figueiredo and P. L. Felmer, A liouville type theorem for Elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. [14] D. G. De Figueiredo and B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Math. Ann., 333 (2005), 231-260. doi: 10.1007/s00208-005-0639-1. [15] B. Gidas and J. Spruk, A priori bounds of positive solutions of nonlinear elliptic equations, Comm. P.D.E., 6 (1981), 883-901. doi: 10.1080/03605308108820196. [16] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle, Commun. Math. Phys., 68 (1979), 525-598. [17] Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $R^N$, Comm. P.D.E., 33 (2008), 263-284. doi: 10.1080/03605300701257476. [18] F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006. [19] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [20] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [21] Y. Li, Remarks on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [22] E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [23] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure and Appl, Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [24] L. Ma and D. Chen, Radial symmetry and monotonicity results for an integral equation, J. Math. Anal. Appl., 2 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064. [25] L. Ma and D. Chen, Radial symmetry and uniqueness of non-negative solutions to an integral system, Math. Comput. Modelling, 49 (2009), 379-385. doi: 10.1016/j.mcm.2008.06.010. [26] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Mathematics, 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [27] 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. [28] E. Mitidieri, Non-existence of positive solutions of semilinear systems in $R^N$, Diff. Int. Eq., 9 (1996), 465-479. [29] W. M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo, Suppl., 8 (1985), 171-185. [30] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system, Diff. Int. Eq., 9 (1996), 635-653. [31] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369-380. [32] J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Comm. P.D.E., 23 (1998), 577-599. doi: 10.1080/03605309808821356. [33] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Advances in Mathematics, 221 (2009), 1409-1437. doi: 10.1016/j.aim.2009.02.014. [34] S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions, Diff. Int. Eq., 8 (1995), 1911-1922. [35] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Eq., 1 (1996), 241-264. [36] X. Yu, Liouville type theorems for integral equations and integral systems, Calculus of Variations and Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.

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##### References:
 [1] H. Berestycki, L. A. Caffarelli and L. Nireberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Norm. Sup. Pisa. C1. Sci., 25 (1997), 69-94. [2] G. Bianchi, Nonexistence of positive solutions to semilinear elliptic equations on $R^N$ or $R^N_+$ through the method of moving planes, Comm. P.D.E., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315. [3] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure App. Math., XLII (1989), 271-297. doi: 10.1002/cpa.3160420304. [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [5] W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [6] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [7] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series, vol. 4, 2010. [8] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [9] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., LIX (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. P.D.E., 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [11] L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86. [12] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Advances in Mathematics, 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [13] D. G. De Figueiredo and P. L. Felmer, A liouville type theorem for Elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. [14] D. G. De Figueiredo and B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Math. Ann., 333 (2005), 231-260. doi: 10.1007/s00208-005-0639-1. [15] B. Gidas and J. Spruk, A priori bounds of positive solutions of nonlinear elliptic equations, Comm. P.D.E., 6 (1981), 883-901. doi: 10.1080/03605308108820196. [16] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle, Commun. Math. Phys., 68 (1979), 525-598. [17] Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $R^N$, Comm. P.D.E., 33 (2008), 263-284. doi: 10.1080/03605300701257476. [18] F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006. [19] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [20] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [21] Y. Li, Remarks on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [22] E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [23] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure and Appl, Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [24] L. Ma and D. Chen, Radial symmetry and monotonicity results for an integral equation, J. Math. Anal. Appl., 2 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064. [25] L. Ma and D. Chen, Radial symmetry and uniqueness of non-negative solutions to an integral system, Math. Comput. Modelling, 49 (2009), 379-385. doi: 10.1016/j.mcm.2008.06.010. [26] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Mathematics, 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [27] 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. [28] E. Mitidieri, Non-existence of positive solutions of semilinear systems in $R^N$, Diff. Int. Eq., 9 (1996), 465-479. [29] W. M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo, Suppl., 8 (1985), 171-185. [30] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system, Diff. Int. Eq., 9 (1996), 635-653. [31] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369-380. [32] J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Comm. P.D.E., 23 (1998), 577-599. doi: 10.1080/03605309808821356. [33] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Advances in Mathematics, 221 (2009), 1409-1437. doi: 10.1016/j.aim.2009.02.014. [34] S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions, Diff. Int. Eq., 8 (1995), 1911-1922. [35] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Eq., 1 (1996), 241-264. [36] X. Yu, Liouville type theorems for integral equations and integral systems, Calculus of Variations and Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.
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