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 & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017
References:
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Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, \emph{Rev. Mat. Iberoamericana}, 20 (2004), 67.   Google Scholar [12] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, \emph{Advances in Mathematics}, 229 (2012), 2835.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar [13] D. G. De Figueiredo and P. L. Felmer, A liouville type theorem for Elliptic systems,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 21 (1994), 387.   Google Scholar [14] D. G. De Figueiredo and B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems,, \emph{Math. Ann.}, 333 (2005), 231.  doi: 10.1007/s00208-005-0639-1.  Google Scholar [15] B. Gidas and J. Spruk, A priori bounds of positive solutions of nonlinear elliptic equations,, \emph{Comm. P.D.E.}, 6 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar [16] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, \emph{Commun. Math. Phys.}, 68 (1979), 525.   Google Scholar [17] Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $R^N$,, \emph{Comm. P.D.E.}, 33 (2008), 263.  doi: 10.1080/03605300701257476.  Google Scholar [18] F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, \emph{Ann. Inst. H. Poincare Anal. Non Lineaire}, 26 (2009), 1.  doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar [19] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar [20] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, \emph{SIAM J. Math. Anal.}, 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar [21] Y. Li, Remarks on some conformally invariant integral equations: the method of moving spheres,, \emph{J. Eur. Math. Soc.}, 6 (2004), 153.   Google Scholar [22] E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar [23] L. Ma and D. Chen, A Liouville type theorem for an integral system,, \emph{Comm. Pure and Appl, 5 (2006), 855.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar [24] L. Ma and D. Chen, Radial symmetry and monotonicity results for an integral equation,, \emph{J. Math. Anal. Appl.}, 2 (2008), 943.  doi: 10.1016/j.jmaa.2007.12.064.  Google Scholar [25] L. Ma and D. Chen, Radial symmetry and uniqueness of non-negative solutions to an integral system,, \emph{Math. Comput. Modelling}, (2009), 379.  doi: 10.1016/j.mcm.2008.06.010.  Google Scholar [26] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Advances in Mathematics}, 226 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar [27] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Ration. Mech. Anal., 195 (2010), 455.  doi: 10.1007/s00205-008-0208-3.  Google Scholar [28] E. Mitidieri, Non-existence of positive solutions of semilinear systems in $R^N$,, \emph{Diff. Int. Eq.}, 9 (1996), 465.   Google Scholar [29] W. M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations,, \emph{Rend. Circ. Mat. Palermo, 8 (1985), 171.   Google Scholar [30] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, \emph{Diff. Int. Eq.}, 9 (1996), 635.   Google Scholar [31] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, \emph{Atti Sem. Mat. Fis. Univ. Modena. Sippl.}, 46 (1998), 369.   Google Scholar [32] J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system,, \emph{Comm. P.D.E.}, 23 (1998), 577.  doi: 10.1080/03605309808821356.  Google Scholar [33] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar [34] S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, \emph{Diff. Int. Eq.}, 8 (1995), 1911.   Google Scholar [35] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, \emph{Adv. Diff. Eq.}, 1 (1996), 241.   Google Scholar [36] X. Yu, Liouville type theorems for integral equations and integral systems,, \emph{Calculus of Variations and Partial Differential Equations}, 46 (2013), 75.  doi: 10.1007/s00526-011-0474-z.  Google Scholar

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References:
 [1] H. Berestycki, L. A. Caffarelli and L. Nireberg, Further qualitative properties for elliptic equations in unbounded domains,, \emph{Ann. Norm. Sup. Pisa. C1. Sci.}, 25 (1997), 69.   Google Scholar [2] G. Bianchi, Nonexistence of positive solutions to semilinear elliptic equations on $R^N$ or $R^N_+$ through the method of moving planes,, \emph{Comm. P.D.E.}, 22 (1997), 1671.  doi: 10.1080/03605309708821315.  Google Scholar [3] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure App. Math.}, XLII (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar [5] W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, \emph{Disc. Cont. Dyn. Sys.}, 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [6] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, \emph{Acta Mathematica Scientia}, 29 (2009), 949.  doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar [7] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series, (2010).   Google Scholar [8] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, \emph{Discrete Contin. Dyn. Syst.}, 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar [9] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure and Appl. Math.}, LIX (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar [10] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. P.D.E.}, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar [11] L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, \emph{Rev. Mat. Iberoamericana}, 20 (2004), 67.   Google Scholar [12] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, \emph{Advances in Mathematics}, 229 (2012), 2835.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar [13] D. G. De Figueiredo and P. L. Felmer, A liouville type theorem for Elliptic systems,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 21 (1994), 387.   Google Scholar [14] D. G. De Figueiredo and B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems,, \emph{Math. Ann.}, 333 (2005), 231.  doi: 10.1007/s00208-005-0639-1.  Google Scholar [15] B. Gidas and J. Spruk, A priori bounds of positive solutions of nonlinear elliptic equations,, \emph{Comm. P.D.E.}, 6 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar [16] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, \emph{Commun. Math. Phys.}, 68 (1979), 525.   Google Scholar [17] Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $R^N$,, \emph{Comm. P.D.E.}, 33 (2008), 263.  doi: 10.1080/03605300701257476.  Google Scholar [18] F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, \emph{Ann. Inst. H. Poincare Anal. Non Lineaire}, 26 (2009), 1.  doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar [19] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar [20] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, \emph{SIAM J. Math. Anal.}, 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar [21] Y. Li, Remarks on some conformally invariant integral equations: the method of moving spheres,, \emph{J. Eur. Math. Soc.}, 6 (2004), 153.   Google Scholar [22] E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar [23] L. Ma and D. Chen, A Liouville type theorem for an integral system,, \emph{Comm. Pure and Appl, 5 (2006), 855.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar [24] L. Ma and D. Chen, Radial symmetry and monotonicity results for an integral equation,, \emph{J. Math. Anal. Appl.}, 2 (2008), 943.  doi: 10.1016/j.jmaa.2007.12.064.  Google Scholar [25] L. Ma and D. Chen, Radial symmetry and uniqueness of non-negative solutions to an integral system,, \emph{Math. Comput. Modelling}, (2009), 379.  doi: 10.1016/j.mcm.2008.06.010.  Google Scholar [26] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Advances in Mathematics}, 226 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar [27] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Ration. Mech. Anal., 195 (2010), 455.  doi: 10.1007/s00205-008-0208-3.  Google Scholar [28] E. Mitidieri, Non-existence of positive solutions of semilinear systems in $R^N$,, \emph{Diff. Int. Eq.}, 9 (1996), 465.   Google Scholar [29] W. M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations,, \emph{Rend. Circ. Mat. Palermo, 8 (1985), 171.   Google Scholar [30] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, \emph{Diff. Int. Eq.}, 9 (1996), 635.   Google Scholar [31] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, \emph{Atti Sem. Mat. Fis. Univ. Modena. Sippl.}, 46 (1998), 369.   Google Scholar [32] J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system,, \emph{Comm. P.D.E.}, 23 (1998), 577.  doi: 10.1080/03605309808821356.  Google Scholar [33] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar [34] S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, \emph{Diff. Int. Eq.}, 8 (1995), 1911.   Google Scholar [35] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, \emph{Adv. Diff. Eq.}, 1 (1996), 241.   Google Scholar [36] X. Yu, Liouville type theorems for integral equations and integral systems,, \emph{Calculus of Variations and Partial Differential Equations}, 46 (2013), 75.  doi: 10.1007/s00526-011-0474-z.  Google Scholar
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