    • 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

 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.
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:
  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  C. Li, Local asymptotic symnwtry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.  Google Scholar  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  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  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  Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Diff. Integ. Eqs., 12 (1999), 601-612. Google Scholar  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  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:
  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  C. Li, Local asymptotic symnwtry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.  Google Scholar  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  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  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  Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Diff. Integ. Eqs., 12 (1999), 601-612. Google Scholar  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  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
  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134  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  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