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Liouville type theorems for singular integral equations and integral systems

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  • 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.
    Mathematics Subject Classification: Primary: 35J60, 45G05; Secondary: 35J47.

    Citation:

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  • [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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