American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations
  • DCDS Home
  • This Issue
  • Next Article
    Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics
January  2015, 35(1): 155-171. doi: 10.3934/dcds.2015.35.155

Liouville theorem for an integral system on the upper half space

Jingbo Dou 1, and  Ye Li 2,

1. 

School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100, China
2. 

Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, United States

Received  January 2014 Revised  June 2014 Published  August 2014

In this paper we establish a Liouville type theorem for an integral system on the upper half space $\mathbb{R}_+^{n}$ \begin{equation*} \begin{cases} u(y)=\int_{\mathbb{R}^{n}_+}\frac{f(v(x))}{|x-y|^{n-\alpha}}dx,&\quad y\in\partial\mathbb{R}^{n}_+,\\ v(x)=\int_{\partial\mathbb{R}^{n}_+}\frac{g(u(y))}{|x-y|^{n-\alpha}}dy,&\quad x\in\mathbb{R}_+^{n}. \end{cases} \end{equation*} This integral system arises from the Euler-Lagrange equation corresponding to Hardy-Littlewood-Sobolev inequality on the upper half space. Under natural structure conditions on $f$ and $g$, we classify positive solutions to the above system basing on the method of moving sphere in integral forms and the Hardy-Littlewood-Sobolev inequality on the upper half space.
Keywords: Liouville type theorem, method of moving sphere., Hardy-Littlewood-Sobolev inequality, integral system.
Mathematics Subject Classification: Primary: 35B53; Secondary: 35B6.
Citation: Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155
References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbbR^n$ or $\mathbbR^n_+$ through the method of moving planes, Comm. Partial Diff. Eqs., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.  Google Scholar

[2]

L. Cao and Z. Dai, A Liouville-type theorem for an integral equation on a half-space $\mathbbR^n_+$, J. Math. Anal. Appl., 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015.  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, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[5]

W. Chen and C. Li, Super Polyharmonic Property of Solutions for PDE Systems and Its Applications, Comm. Pure and Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.  Google Scholar

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[7]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[8]

C. Chen and C. S. Lin, Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent, Duke Math. J., 78 (1995), 315-334. doi: 10.1215/S0012-7094-95-07814-4.  Google Scholar

[9]

L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86.  Google Scholar

[10]

J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math., 54 (2011), 753-768. doi: 10.1007/s11425-011-4177-x.  Google Scholar

[11]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Notices, 2014 (2014), 37 pp. doi: 10.1093/imrn/rnt213.  Google Scholar

[12]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, in Math. Anal. Appl., Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.  Google Scholar

[13]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

[14]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Eqs., 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[15]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[16]

F. B. 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

[17]

C. Li, Local asymptotic symnwtry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.  Google Scholar

[18]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.  Google Scholar

[19]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. D'Anal. Math., 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[20]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[21]

Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Diff. Integ. Eqs., 12 (1999), 601-612.  Google Scholar

[22]

W. Reichel and T. Weth, A prior bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827. doi: 10.1007/s00209-008-0352-3.  Google Scholar

[23]

X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. PDE, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

show all references

References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbbR^n$ or $\mathbbR^n_+$ through the method of moving planes, Comm. Partial Diff. Eqs., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.  Google Scholar

[2]

L. Cao and Z. Dai, A Liouville-type theorem for an integral equation on a half-space $\mathbbR^n_+$, J. Math. Anal. Appl., 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015.  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, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[5]

W. Chen and C. Li, Super Polyharmonic Property of Solutions for PDE Systems and Its Applications, Comm. Pure and Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.  Google Scholar

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[7]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[8]

C. Chen and C. S. Lin, Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent, Duke Math. J., 78 (1995), 315-334. doi: 10.1215/S0012-7094-95-07814-4.  Google Scholar

[9]

L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86.  Google Scholar

[10]

J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math., 54 (2011), 753-768. doi: 10.1007/s11425-011-4177-x.  Google Scholar

[11]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Notices, 2014 (2014), 37 pp. doi: 10.1093/imrn/rnt213.  Google Scholar

[12]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, in Math. Anal. Appl., Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.  Google Scholar

[13]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

[14]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Eqs., 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[15]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[16]

F. B. 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

[17]

C. Li, Local asymptotic symnwtry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.  Google Scholar

[18]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.  Google Scholar

[19]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. D'Anal. Math., 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[20]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[21]

Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Diff. Integ. Eqs., 12 (1999), 601-612.  Google Scholar

[22]

W. Reichel and T. Weth, A prior bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827. doi: 10.1007/s00209-008-0352-3.  Google Scholar

[23]

X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. PDE, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

[1]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
[2]

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
[3]

Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027
[4]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
[5]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[6]

Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653
[7]

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
[8]

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
[9]

Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057
[10]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[11]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[12]

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 & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
[13]

Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022
[14]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511
[15]

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 & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[16]

Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure & Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008
[17]

Minbo Yang, Fukun Zhao, Shunneng Zhao. Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5209-5241. doi: 10.3934/dcds.2021074
[18]

Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021134
[19]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[20]

Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences