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

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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 and Continuous Dynamical Systems, 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.
[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.

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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