# American Institute of Mathematical Sciences

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

## An extended discrete Hardy-Littlewood-Sobolev inequality

 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.
Citation: Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete and 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. [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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##### 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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