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May  2014, 34(5): 1951-1959. doi: 10.3934/dcds.2014.34.1951

An extended discrete Hardy-Littlewood-Sobolev inequality

Ze Cheng 1, and  Congming Li 1,

1. 

Department of Applied Mathematics, University of Colorado at Boulder, Colorado, United States, United States

Received  May 2013 Revised  July 2013 Published  October 2013

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the ``critical'' case: $μ=n$. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: $μ=n$ and $p=q$, by limiting the inequality on a finite domain. The best constant in the inequality and its corresponding solution, the optimizer, are studied. First, we obtain a sharp estimate for the best constant. Then for the optimizer, we prove the uniqueness and a symmetry property. This is achieved by proving that the corresponding Euler-Lagrange equation has a unique nontrivial nonnegative critical point. Also, by using a discrete version of maximum principle, we prove certain monotonicity of this optimizer.
Keywords: HLS inequality, Euler-Lagrange equation., maximum principle.
Mathematics Subject Classification: Primary: 35A2.
Citation: Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm Pure Appl Anal, 4 (2005), 1-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. & Cont. Dynamics Sys., 12 (2005), 347-354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge at the University Press, 1952.  Google Scholar

[16]

J. Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143. Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[22]

O. Perron, Zur Theorie der Matrices, Mathematische Annalen, 64 (1907), 248-263. doi: 10.1007/BF01449896.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm Pure Appl Anal, 4 (2005), 1-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. & Cont. Dynamics Sys., 12 (2005), 347-354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge at the University Press, 1952.  Google Scholar

[16]

J. Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143. Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[22]

O. Perron, Zur Theorie der Matrices, Mathematische Annalen, 64 (1907), 248-263. doi: 10.1007/BF01449896.  Google Scholar

[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.  Google Scholar

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