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January  2020, 40(1): 467-489. doi: 10.3934/dcds.2020018

Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type

Xiaoqian Liu 1, and  Yutian Lei 2,,

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

* Corresponding author: Yutian Lei

Received  March 2019 Published  October 2019

Fund Project: This research is supported by the National Natural Science Foundation of China (11871278, 11671209).

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.

Keywords: Weighted Hardy-Littlewood-Sobolev inequality, integral system, existence of positive solution, Serrin-type condition.
Mathematics Subject Classification: 45E10, 45G05, 45M20.
Citation: Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
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.  Google Scholar

[2]

W. Beckner, Pitt's inequality and the uncertainty principle, Proc. Amer. Math. Soc., 123 (1995), 1897-1905.  doi: 10.2307/2161009.  Google Scholar

[3]

W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math., 20 (2008), 587-606.  doi: 10.1515/FORUM.2008.030.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

W. Beckner, Pitt's inequality and the uncertainty principle, Proc. Amer. Math. Soc., 123 (1995), 1897-1905.  doi: 10.2307/2161009.  Google Scholar

[3]

W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math., 20 (2008), 587-606.  doi: 10.1515/FORUM.2008.030.  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), 27-67.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princetion Math. Series, Vol. 30, Princetion University Press, Princetion, NJ, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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