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Anosov diffeomorphism with a horseshoe that attracts almost any point
Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type
1. | Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China |
2. | Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China |
This paper is concerned with the existence/nonexistence of positive solutions of a weighted Hardy-Littlewood-Sobolev type integral system. Such a system is related to the extremal functions of the weighted Hardy-Littlewood-Sobolev inequality. The Serrin-type condition is critical for existence of positive solutions in $ L_{loc}^\infty(R^n \setminus \{0\}) $. When the Serrin-type condition does not hold, we prove the nonexistence by an iteration process. In addition, we find three pairs of radial solutions when the Serrin-type condition holds. One is singular, and the other two are integrable in $ R^n $ and decaying fast and slowly respectively.
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A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.
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W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Discrete Contin. Dyn. Syst., (2005), 164–172. |
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W. Chen and C. Li,
The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.
doi: 10.1090/S0002-9939-07-09232-5. |
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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. |
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L. D'Ambrosio and E. Mitidieri,
Hardy-Littlewood-Sobolev systems and related Liouville theorems, Discrete Contin. Dyn. Syst. Series S, 7 (2014), 653-671.
doi: 10.3934/dcdss.2014.7.653. |
[11] |
F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington, NC, USA, 2002,327–335. |
[12] |
J. Hulshof and R. C. A. M. Van der Vorst,
Asymptotic behavior of ground states, Proc. Amer. Math. Soc., 124 (1996), 2423-2431.
doi: 10.1090/S0002-9939-96-03669-6. |
[13] |
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. |
[14] |
C. Jin and C. Li,
Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[15] |
Y. Lei,
Critical conditions and finite energy solutions of several nonlinear elliptic PDEs in $R^n$, J. Differential Equations, 258 (2015), 4033-4061.
doi: 10.1016/j.jde.2015.01.043. |
[16] |
Y. Lei and C. Li,
Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.
doi: 10.3934/dcds.2016.36.3277. |
[17] |
Y. Lei, C. Li and C. Ma,
Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[18] |
Y. Lei and Z. Lü,
Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 33 (2013), 1987-2005.
doi: 10.3934/dcds.2013.33.1987. |
[19] |
Y. Lei and C. Ma,
Asymptotic behavior for solutions of some integral equations, Commun. Pure Appl. Anal., 10 (2011), 193-207.
doi: 10.3934/cpaa.2011.10.193. |
[20] |
C. Li and J. Lim,
The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal., 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[21] |
E. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[22] |
Y. Lü and Z. Lü,
Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system, Discrete Contin. Dyn. Syst., 36 (2016), 3791-3810.
doi: 10.3934/dcds.2016.36.3791. |
[23] |
E. Mitidieri and S. Pohozaev,
A priori estimates and blow-up solutions to nonlinear partail differential equations and inequalities, Proc. Steklov Institue Maths., 234 (2001), 1-384.
|
[24] |
M. Onodera,
On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510.
doi: 10.1016/j.jmaa.2011.12.004. |
[25] |
M. A. S. Souto,
A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258.
|
[26] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. |
[27] |
E. Stein and G. Weiss,
Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
doi: 10.1512/iumj.1958.7.57030. |
[28] |
D. Wu, Z. Shi and D. Yan,
Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality, Sci. China Math., 57 (2014), 963-970.
doi: 10.1007/s11425-013-4681-2. |
[29] |
Y. Zhao,
Regularity and symmetry for solutions to a system of weighted integral equations, J. Math. Anal. Appl., 391 (2012), 209-222.
doi: 10.1016/j.jmaa.2012.02.016. |
show all references
References:
[1] |
J. Bebernes, Y. Lei and C. Li,
A singularity analysis of positive solutions to an Euler-Lagrange integral system, Rocky Mountain J. Math., 41 (2011), 387-410.
doi: 10.1216/RMJ-2011-41-2-387. |
[2] |
W. Beckner,
Pitt's inequality and the uncertainty principle, Proc. Amer. Math. Soc., 123 (1995), 1897-1905.
doi: 10.2307/2161009. |
[3] |
W. Beckner,
Weighted inequalities and Stein-Weiss potentials, Forum Math., 20 (2008), 587-606.
doi: 10.1515/FORUM.2008.030. |
[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] |
D. Chen and L. Ma,
A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.
doi: 10.3934/cpaa.2006.5.855. |
[6] |
W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Discrete Contin. Dyn. Syst., (2005), 164–172. |
[7] |
W. Chen and C. Li,
The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.
doi: 10.1090/S0002-9939-07-09232-5. |
[8] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., 2010. |
[9] |
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. |
[10] |
L. D'Ambrosio and E. Mitidieri,
Hardy-Littlewood-Sobolev systems and related Liouville theorems, Discrete Contin. Dyn. Syst. Series S, 7 (2014), 653-671.
doi: 10.3934/dcdss.2014.7.653. |
[11] |
F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington, NC, USA, 2002,327–335. |
[12] |
J. Hulshof and R. C. A. M. Van der Vorst,
Asymptotic behavior of ground states, Proc. Amer. Math. Soc., 124 (1996), 2423-2431.
doi: 10.1090/S0002-9939-96-03669-6. |
[13] |
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. |
[14] |
C. Jin and C. Li,
Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[15] |
Y. Lei,
Critical conditions and finite energy solutions of several nonlinear elliptic PDEs in $R^n$, J. Differential Equations, 258 (2015), 4033-4061.
doi: 10.1016/j.jde.2015.01.043. |
[16] |
Y. Lei and C. Li,
Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.
doi: 10.3934/dcds.2016.36.3277. |
[17] |
Y. Lei, C. Li and C. Ma,
Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61.
doi: 10.1007/s00526-011-0450-7. |
[18] |
Y. Lei and Z. Lü,
Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 33 (2013), 1987-2005.
doi: 10.3934/dcds.2013.33.1987. |
[19] |
Y. Lei and C. Ma,
Asymptotic behavior for solutions of some integral equations, Commun. Pure Appl. Anal., 10 (2011), 193-207.
doi: 10.3934/cpaa.2011.10.193. |
[20] |
C. Li and J. Lim,
The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal., 6 (2007), 453-464.
doi: 10.3934/cpaa.2007.6.453. |
[21] |
E. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[22] |
Y. Lü and Z. Lü,
Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system, Discrete Contin. Dyn. Syst., 36 (2016), 3791-3810.
doi: 10.3934/dcds.2016.36.3791. |
[23] |
E. Mitidieri and S. Pohozaev,
A priori estimates and blow-up solutions to nonlinear partail differential equations and inequalities, Proc. Steklov Institue Maths., 234 (2001), 1-384.
|
[24] |
M. Onodera,
On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510.
doi: 10.1016/j.jmaa.2011.12.004. |
[25] |
M. A. S. Souto,
A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258.
|
[26] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970. |
[27] |
E. Stein and G. Weiss,
Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
doi: 10.1512/iumj.1958.7.57030. |
[28] |
D. Wu, Z. Shi and D. Yan,
Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality, Sci. China Math., 57 (2014), 963-970.
doi: 10.1007/s11425-013-4681-2. |
[29] |
Y. Zhao,
Regularity and symmetry for solutions to a system of weighted integral equations, J. Math. Anal. Appl., 391 (2012), 209-222.
doi: 10.1016/j.jmaa.2012.02.016. |
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