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An extended discrete Hardy-Littlewood-Sobolev inequality
1. | Department of Applied Mathematics, University of Colorado at Boulder, Colorado, United States, United States |
References:
[1] |
W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev Inequalities and Systems of Integral Equations, Discrete and Continuous Dynamical Systems, (2005), 164-172. |
[2] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[3] |
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. |
[4] |
W. Chen and C. Li, Indefinite elliptic problems in a domain, Discrete Contin. Dynam. Systems, 3 (1997), 333-340.
doi: 10.3934/dcds.1997.3.333. |
[5] |
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. |
[6] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm Pure Appl Anal, 4 (2005), 1-8. |
[7] |
W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. AMS, 136 (2008), 955-962.
doi: 10.1090/S0002-9939-07-09232-5. |
[8] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. in Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[9] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. & Cont. Dynamics Sys., 12 (2005), 347-354. |
[10] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[11] |
L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Mathematics, 128. Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511569203. |
[12] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $R^n$, Mathematical Analysis and Applications, Part A, pp. 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[13] |
F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res Lett, 14 (2007), 373-383.
doi: 10.4310/MRL.2007.v14.n3.a2. |
[14] |
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. |
[15] |
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge at the University Press, 1952. |
[16] |
J. Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143. Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
[17] |
Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[18] |
Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558.
doi: 10.3934/dcds.2011.30.547. |
[19] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal, 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[20] |
C. Li and J. Villavert, An extension of the Hardy-Littlewood-Pólya inequality, Acta Mathematica Scientia, 31 (2011), 2285-2288.
doi: 10.1016/S0252-9602(11)60400-1. |
[21] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[22] |
O. Perron, Zur Theorie der Matrices, Mathematische Annalen, 64 (1907), 248-263.
doi: 10.1007/BF01449896. |
[23] |
E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
doi: 10.1512/iumj.1958.7.57030. |
show all references
References:
[1] |
W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev Inequalities and Systems of Integral Equations, Discrete and Continuous Dynamical Systems, (2005), 164-172. |
[2] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[3] |
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. |
[4] |
W. Chen and C. Li, Indefinite elliptic problems in a domain, Discrete Contin. Dynam. Systems, 3 (1997), 333-340.
doi: 10.3934/dcds.1997.3.333. |
[5] |
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. |
[6] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm Pure Appl Anal, 4 (2005), 1-8. |
[7] |
W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. AMS, 136 (2008), 955-962.
doi: 10.1090/S0002-9939-07-09232-5. |
[8] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. in Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[9] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. & Cont. Dynamics Sys., 12 (2005), 347-354. |
[10] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[11] |
L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Mathematics, 128. Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511569203. |
[12] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $R^n$, Mathematical Analysis and Applications, Part A, pp. 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[13] |
F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res Lett, 14 (2007), 373-383.
doi: 10.4310/MRL.2007.v14.n3.a2. |
[14] |
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. |
[15] |
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge at the University Press, 1952. |
[16] |
J. Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143. Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
[17] |
Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[18] |
Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation, Discrete Contin. Dyn. Syst., 30 (2011), 547-558.
doi: 10.3934/dcds.2011.30.547. |
[19] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal, 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[20] |
C. Li and J. Villavert, An extension of the Hardy-Littlewood-Pólya inequality, Acta Mathematica Scientia, 31 (2011), 2285-2288.
doi: 10.1016/S0252-9602(11)60400-1. |
[21] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[22] |
O. Perron, Zur Theorie der Matrices, Mathematische Annalen, 64 (1907), 248-263.
doi: 10.1007/BF01449896. |
[23] |
E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
doi: 10.1512/iumj.1958.7.57030. |
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