Hardy-Littlewood-Sobolev systems and related Liouville theorems

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August 2014, 7(4): 653-671. doi: 10.3934/dcdss.2014.7.653

Hardy-Littlewood-Sobolev systems and related Liouville theorems

Lorenzo D'Ambrosio ^1, and Enzo Mitidieri ^2,

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

Received November 2013 Published February 2014

Abstract
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We prove some Liouville theorems for systems of integral equations and inequalities related to weighted Hardy-Littlewood-Sobolev inequality type on $R^N$ . Some semilinear singular or degenerate higher order elliptic inequalities associated to polyharmonic operators are considered. Special cases include the Hénon-Lane-Emden system.

Keywords: Liouville theorems, Representation formulae, Hardy-Hénon-Emdem system., Hardy-Littlewood-Sobolev inequality.

Mathematics Subject Classification: Primary: 35C15; Secondary: 35J48, 45G1.

Citation: 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

References:

[1]	H. Brezis and and S. Kamin, Sublinear elliptic equations in $R^N$,, Manuscripta Math., 74 (1992), 87. doi: 10.1007/BF02567660. Google Scholar
[2]	A. Björn and J. Biörn, Nonlinear Potential Theory on Metric Spaces,, EMS Tracts in Mathematics, (2011). doi: 10.4171/099. Google Scholar
[3]	G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville Theorems for some nonlinear inequalities,, Proc. Steklov Inst. Math., 260 (2008), 90. doi: 10.1134/S0081543808010070. Google Scholar
[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. doi: 10.1007/s00032-008-0090-3. Google Scholar
[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. Google Scholar
[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. doi: 10.1081/PDE-200044445. Google Scholar
[7]	W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4 (2005), 1. Google Scholar
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar
[9]	C. Cowan, A Liouville theorem for a fourth order Hénon equation,, , (). Google Scholar
[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. doi: 10.1090/S0002-9947-05-03717-7. Google Scholar
[11]	L. Euler, Specimen transformationis singularis serierum,, Nova Acta Acad. Petropol., 7 (1778), 58. Google Scholar
[12]	M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture,, , (). Google Scholar
[13]	J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford University Press, (1993). Google Scholar
[14]	W. K. Hayman and P. B. Kennedy, Subharmonic functions,, I, (1976). Google Scholar
[15]	C. Jin and C. Li, Qualitative Analysis of Some Systems of Integral Equations,, Cal. Var. PDEs, 26 (2006), 447. doi: 10.1007/s00526-006-0013-5. Google Scholar
[16]	C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X. Google Scholar
[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. Google Scholar
[18]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorem for polyharmonic systems in $R^N$,, J. Differential Eq., 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016. Google Scholar
[19]	E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in $R^N$,, Differential & Integral Eq., 9 (1996), 465. Google Scholar
[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. Google Scholar
[21]	E. M. Stein and G. Weiss, Fractional Integrals in n-dimensional Euclidean space,, J. Math. Mech., 7 (1958). Google Scholar
[22]	X. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

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References:

[1]	H. Brezis and and S. Kamin, Sublinear elliptic equations in $R^N$,, Manuscripta Math., 74 (1992), 87. doi: 10.1007/BF02567660. Google Scholar
[2]	A. Björn and J. Biörn, Nonlinear Potential Theory on Metric Spaces,, EMS Tracts in Mathematics, (2011). doi: 10.4171/099. Google Scholar
[3]	G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville Theorems for some nonlinear inequalities,, Proc. Steklov Inst. Math., 260 (2008), 90. doi: 10.1134/S0081543808010070. Google Scholar
[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. doi: 10.1007/s00032-008-0090-3. Google Scholar
[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. Google Scholar
[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. doi: 10.1081/PDE-200044445. Google Scholar
[7]	W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4 (2005), 1. Google Scholar
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar
[9]	C. Cowan, A Liouville theorem for a fourth order Hénon equation,, , (). Google Scholar
[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. doi: 10.1090/S0002-9947-05-03717-7. Google Scholar
[11]	L. Euler, Specimen transformationis singularis serierum,, Nova Acta Acad. Petropol., 7 (1778), 58. Google Scholar
[12]	M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture,, , (). Google Scholar
[13]	J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford University Press, (1993). Google Scholar
[14]	W. K. Hayman and P. B. Kennedy, Subharmonic functions,, I, (1976). Google Scholar
[15]	C. Jin and C. Li, Qualitative Analysis of Some Systems of Integral Equations,, Cal. Var. PDEs, 26 (2006), 447. doi: 10.1007/s00526-006-0013-5. Google Scholar
[16]	C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X. Google Scholar
[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. Google Scholar
[18]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorem for polyharmonic systems in $R^N$,, J. Differential Eq., 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016. Google Scholar
[19]	E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in $R^N$,, Differential & Integral Eq., 9 (1996), 465. Google Scholar
[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. Google Scholar
[21]	E. M. Stein and G. Weiss, Fractional Integrals in n-dimensional Euclidean space,, J. Math. Mech., 7 (1958). Google Scholar
[22]	X. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

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