-
Previous Article
On two phase free boundary problems governed by elliptic equations with distributed sources
- DCDS-S Home
- This Issue
-
Next Article
An excess-decay result for a class of degenerate elliptic equations
Hardy-Littlewood-Sobolev systems and related Liouville theorems
1. | Dipartimento di Matematica, Università degli Studi di Bari, via E.Orabona 4, I-70125 Bari, Italy |
2. | Dipartimento di Matematica e Geoscienze, Università di Trieste, via Alfonso Valerio 12/1, I-34100 Trieste, Italy |
References:
[1] |
H. Brezis and and S. Kamin, Sublinear elliptic equations in $R^N$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[2] |
A. Björn and J. Biörn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zürich, 2011.
doi: 10.4171/099. |
[3] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville Theorems for some nonlinear inequalities, Proc. Steklov Inst. Math., 260 (2008), 90-111.
doi: 10.1134/S0081543808010070. |
[4] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3. |
[5] |
W. Chen, C. Jin, C. Li and Jisun Lim, Weighted Hardy-Littlewood-Sobolev inequalities and Systems of integral equations, Disc. and Cont. Dynamics Sys. Supplement, (2005), 164-173. |
[6] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[7] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 1-8. |
[8] |
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. |
[9] |
C. Cowan, A Liouville theorem for a fourth order Hénon equation, arXiv:1110.2246. |
[10] |
L. D'Ambrosio, E. Mitidieri and S. I. Pohozaev, Representation formulae and inequalities for solutions of a class of second order partial differential equations, Trans. Amer. Math. Soc., 358 (2005), 893-910.
doi: 10.1090/S0002-9947-05-03717-7. |
[11] |
L. Euler, Specimen transformationis singularis serierum, Nova Acta Acad. Petropol., 7 (1778), 58-70. |
[12] |
M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, arXiv:1107.561. |
[13] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993. |
[14] |
W. K. Hayman and P. B. Kennedy, Subharmonic functions, I, Academic Press, London, New York, San Francisco, 1976. |
[15] |
C. Jin and C. Li, Qualitative Analysis of Some Systems of Integral Equations, Cal. Var. PDEs, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[16] |
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. |
[17] |
Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, Journal of the European Mathematical Society, 6 (2004), 153-180. |
[18] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorem for polyharmonic systems in $R^N$, J. Differential Eq., 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[19] |
E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in $R^N$, Differential & Integral Eq., 9 (1996), 465-479. |
[20] |
E. Mitidieri and S. I. Pohozaev, A priori estimates and nonexistence of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-375. |
[21] |
E. M. Stein and G. Weiss, Fractional Integrals in n-dimensional Euclidean space, J. Math. Mech., 7 (1958). |
[22] |
X. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
show all references
References:
[1] |
H. Brezis and and S. Kamin, Sublinear elliptic equations in $R^N$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[2] |
A. Björn and J. Biörn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zürich, 2011.
doi: 10.4171/099. |
[3] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville Theorems for some nonlinear inequalities, Proc. Steklov Inst. Math., 260 (2008), 90-111.
doi: 10.1134/S0081543808010070. |
[4] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3. |
[5] |
W. Chen, C. Jin, C. Li and Jisun Lim, Weighted Hardy-Littlewood-Sobolev inequalities and Systems of integral equations, Disc. and Cont. Dynamics Sys. Supplement, (2005), 164-173. |
[6] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[7] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 1-8. |
[8] |
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. |
[9] |
C. Cowan, A Liouville theorem for a fourth order Hénon equation, arXiv:1110.2246. |
[10] |
L. D'Ambrosio, E. Mitidieri and S. I. Pohozaev, Representation formulae and inequalities for solutions of a class of second order partial differential equations, Trans. Amer. Math. Soc., 358 (2005), 893-910.
doi: 10.1090/S0002-9947-05-03717-7. |
[11] |
L. Euler, Specimen transformationis singularis serierum, Nova Acta Acad. Petropol., 7 (1778), 58-70. |
[12] |
M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, arXiv:1107.561. |
[13] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993. |
[14] |
W. K. Hayman and P. B. Kennedy, Subharmonic functions, I, Academic Press, London, New York, San Francisco, 1976. |
[15] |
C. Jin and C. Li, Qualitative Analysis of Some Systems of Integral Equations, Cal. Var. PDEs, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[16] |
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. |
[17] |
Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, Journal of the European Mathematical Society, 6 (2004), 153-180. |
[18] |
J. Liu, Y. Guo and Y. Zhang, Liouville-type theorem for polyharmonic systems in $R^N$, J. Differential Eq., 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[19] |
E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in $R^N$, Differential & Integral Eq., 9 (1996), 465-479. |
[20] |
E. Mitidieri and S. I. Pohozaev, A priori estimates and nonexistence of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-375. |
[21] |
E. M. Stein and G. Weiss, Fractional Integrals in n-dimensional Euclidean space, J. Math. Mech., 7 (1958). |
[22] |
X. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
[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] |
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 |
[3] |
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 |
[4] |
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 |
[5] |
Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134 |
[6] |
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 |
[7] |
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 |
[8] |
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 |
[9] |
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 |
[10] |
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 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
Kods Hassine. Existence and uniqueness of radial solutions for Hardy-Hénon equations involving k-Hessian operators. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022084 |
[17] |
Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032 |
[18] |
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 |
[19] |
Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915 |
[20] |
Rong Zhang. Nonexistence of Positive Solutions for high-order Hardy-H$ \acute{e} $non Systems on $ \mathbb{R}^{n} $. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2857-2872. doi: 10.3934/cpaa.2022078 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]