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

Previous Article
Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials
DCDS Home
This Issue
Next Article
The initial-boundary value problem for the compressible viscoelastic flows

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

Abstract
Related Papers

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 and 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.
[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.
[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.
[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.
[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, Commun. Pure Appl. Anal., 4 (2005), 1-8. doi: 10.3934/cpaa.2005.4.1.
[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.
[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.
[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.
[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.
[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.
[12]	G. Hardy, J. Littlewood and J. Pólya, Inequalities, $2^{nd}$ edition, Cambridge University Press, 1952.
[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.
[14]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Math., 118 (1983), 349-374. doi: 10.2307/2007032.

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.
[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.
[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.
[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.
[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, Commun. Pure Appl. Anal., 4 (2005), 1-8. doi: 10.3934/cpaa.2005.4.1.
[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.
[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.
[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.
[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.
[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.
[12]	G. Hardy, J. Littlewood and J. Pólya, Inequalities, $2^{nd}$ edition, Cambridge University Press, 1952.
[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.
[14]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Math., 118 (1983), 349-374. doi: 10.2307/2007032.

[1]	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
[2]	Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057
[3]	Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027
[4]	Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[5]	Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[6]	Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure and Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[7]	Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
[8]	Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653
[9]	Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[10]	Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022
[11]	Jianqing Chen. Best constant of 3D Anisotropic Sobolev inequality and its applications. Communications on Pure and Applied Analysis, 2010, 9 (3) : 655-666. doi: 10.3934/cpaa.2010.9.655
[12]	José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[13]	Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure and Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
[14]	Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
[15]	Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure and Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008
[16]	Minbo Yang, Fukun Zhao, Shunneng Zhao. Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5209-5241. doi: 10.3934/dcds.2021074
[17]	Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223
[18]	Julián Fernández Bonder, Julio D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Communications on Pure and Applied Analysis, 2002, 1 (3) : 359-378. doi: 10.3934/cpaa.2002.1.359
[19]	Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
[20]	John Villavert. Sharp existence criteria for positive solutions of Hardy--Sobolev type systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 493-515. doi: 10.3934/cpaa.2015.14.493

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Genggeng Huang Congming Li Ximing Yin
on Google Scholar
Genggeng Huang Congming Li Ximing Yin

[Back to Top]