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March  2015, 35(3): 935-942. doi: 10.3934/dcds.2015.35.935

Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality

Genggeng Huang 1, , Congming Li 1, and  Ximing Yin 1,

1. 

Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China, China, China

Received  February 2014 Revised  April 2014 Published  October 2014

In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \begin{equation} \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-\alpha}}\leq C_{r,s,\alpha} |f|_{l^r} |g|_{l^s}, \end{equation}where $i,j\in \mathbb Z^n$, $r,s>1$, $0 < \alpha < n$, and $\frac {1} {r} + \frac {1} {s} + \frac {n-\alpha}{n} \geq 2$. Indeed, we prove that the best constant is attainable in the supercritical case $\frac {1}{r} + \frac {1} {s} + \frac {n-\alpha}{n} > 2$.
Keywords: concentration compactness, supercritical., Discrete Hardy-Littlewood-Sobolev inequality, best constant, existence.
Mathematics Subject Classification: Primary: 35A2.
Citation: Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
References:
[1]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society, 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[2]

X. Chen and X. Zhen, Optimal summation interval and nonexistence of positive solutions to a discrete sytem, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 1720-1730. doi: 10.1016/S0252-9602(14)60117-X.  Google Scholar

[3]

W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Discrete Contin. Dyn. Syst. 2005, suppl., (2005), 164-172.  Google Scholar

[4]

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

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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, Commun. Pure Appl. Anal., 4 (2005), 1-8. doi: 10.3934/cpaa.2005.4.1.  Google Scholar

[7]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soci., 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, Communications in Partial Difference 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, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347.  Google Scholar

[10]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Communications on pure and applied mathematics, 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[11]

Z. Cheng and C. Li, An extended discrete Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 34 (2014), 1951-1959. doi: 10.3934/dcds.2014.34.1951.  Google Scholar

[12]

G. Hardy, J. Littlewood and J. Pólya, Inequalities, $2^{nd}$ edition, Cambridge University Press, 1952.  Google Scholar

[13]

C. Li and J. Villavert, An extention of the Hardy-Littlewood-Pólya inequality, Acta Math. Scientia, 31 (2011), 2285-2288. doi: 10.1016/S0252-9602(11)60400-1.  Google Scholar

[14]

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

show all references

References:
[1]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society, 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[2]

X. Chen and X. Zhen, Optimal summation interval and nonexistence of positive solutions to a discrete sytem, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 1720-1730. doi: 10.1016/S0252-9602(14)60117-X.  Google Scholar

[3]

W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Discrete Contin. Dyn. Syst. 2005, suppl., (2005), 164-172.  Google Scholar

[4]

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

[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, Commun. Pure Appl. Anal., 4 (2005), 1-8. doi: 10.3934/cpaa.2005.4.1.  Google Scholar

[7]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soci., 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, Communications in Partial Difference 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, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347.  Google Scholar

[10]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Communications on pure and applied mathematics, 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[11]

Z. Cheng and C. Li, An extended discrete Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 34 (2014), 1951-1959. doi: 10.3934/dcds.2014.34.1951.  Google Scholar

[12]

G. Hardy, J. Littlewood and J. Pólya, Inequalities, $2^{nd}$ edition, Cambridge University Press, 1952.  Google Scholar

[13]

C. Li and J. Villavert, An extention of the Hardy-Littlewood-Pólya inequality, Acta Math. Scientia, 31 (2011), 2285-2288. doi: 10.1016/S0252-9602(11)60400-1.  Google Scholar

[14]

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

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