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

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.

Citation: 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
References:
[1]

J. BebernesY. 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. CaristiL. 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. ChenC. 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. LeiC. 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. WuZ. 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. BebernesY. 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. CaristiL. 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. ChenC. 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. LeiC. 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. WuZ. 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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