Sharp existence criteria for positive solutions of Hardy--Sobolev type systems

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March 2015, 14(2): 493-515. doi: 10.3934/cpaa.2015.14.493

Sharp existence criteria for positive solutions of Hardy--Sobolev type systems

John Villavert ^1,

1.	Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, China

Received February 2014 Revised September 2014 Published December 2014

Abstract
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This paper examines systems of poly-harmonic equations of the Hardy--Sobolev type and the closely related weighted systems of integral equations involving Riesz potentials. Namely, it is shown that the two systems are equivalent under some appropriate conditions. Then a sharp criterion for the existence and non-existence of positive solutions is determined for both differential and integral versions of a Hardy--Sobolev type system with variable coefficients. In the constant coefficient case, Liouville type theorems for positive radial solutions are also established using radial decay estimates and Pohozaev type identities in integral form.

Keywords: Hardy--Littlewood--Sobolev inequality, Lane--Emden equations, Hardy--Sobolev inequality, fractional integrals, poly-harmonic equations..

Mathematics Subject Classification: Primary: 35B53, 45G05, 45G15; Secondary: 35J48, 35J9.

Citation: John Villavert. Sharp existence criteria for positive solutions of Hardy--Sobolev type systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 493-515. doi: 10.3934/cpaa.2015.14.493

References:

[1]	J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana Univ. Math. J., 51 (2002), 37-51.
[2]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.
[3]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math., 53 (1984), 259-275.
[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]	G. Caristi, E. Mitidieri and R. Soranzo, Isolated singularities of polyharmonic equations, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 257-294.
[6]	F. Catrina and Z. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.
[7]	W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems on a half space, J. Funct. Anal., 265 (2013), 1522-1555. doi: 10.1016/j.jfa.2013.06.010.
[8]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.
[9]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.
[10]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.
[11]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
[12]	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.
[13]	L. D'Ambrosio and E. Mitidieri, Hardy-Littlewood-Sobolev systems and related Liouville theorems, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 653-671. doi: 10.3934/dcdss.2014.7.653.
[14]	D. G. De Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21 (1994), 387-397.
[15]	B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Adv. Math. Suppl. Studies A, 7 (1981), 369-402.
[16]	B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.
[17]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.
[18]	G. H. Hardy, J. E. Littlewood and G. Pólya, Some properties of fractional integral (1), Math. Z., 27 (1928), 565-606. doi: 10.1007/BF01171116.
[19]	M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24 (1973), 229-238.
[20]	Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 253 (2013), 1774-1799. doi: 10.1016/j.jde.2012.11.008.
[21]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, preprint, arXiv:1301.6235.
[22]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Ivent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.
[23]	C. Li, A degree theory approach for the shooting method, preprint, arXiv:1301.6232.
[24]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.
[25]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270.
[26]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^N$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.
[27]	E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.
[28]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differ. Integral Equations, 9 (1996), 465-480.
[29]	Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon systems, Adv. Differential Equations, 17 (2012), 605-634.
[30]	Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differential Equations, 252 (2012), 2544-2562. doi: 10.1016/j.jde.2011.09.022.
[31]	P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8.
[32]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653.
[33]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.
[34]	E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
[35]	J. Villavert, Shooting with degree theory: Analysis of some weighted poly-harmonic systems, J. Differential Equations, 257 (2014), 1148-1167. doi: 10.1016/j.jde.2014.05.003.
[36]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.

show all references

References:

[1]	J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana Univ. Math. J., 51 (2002), 37-51.
[2]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.
[3]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math., 53 (1984), 259-275.
[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]	G. Caristi, E. Mitidieri and R. Soranzo, Isolated singularities of polyharmonic equations, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 257-294.
[6]	F. Catrina and Z. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.
[7]	W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems on a half space, J. Funct. Anal., 265 (2013), 1522-1555. doi: 10.1016/j.jfa.2013.06.010.
[8]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.
[9]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.
[10]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.
[11]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
[12]	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.
[13]	L. D'Ambrosio and E. Mitidieri, Hardy-Littlewood-Sobolev systems and related Liouville theorems, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 653-671. doi: 10.3934/dcdss.2014.7.653.
[14]	D. G. De Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21 (1994), 387-397.
[15]	B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Adv. Math. Suppl. Studies A, 7 (1981), 369-402.
[16]	B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.
[17]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.
[18]	G. H. Hardy, J. E. Littlewood and G. Pólya, Some properties of fractional integral (1), Math. Z., 27 (1928), 565-606. doi: 10.1007/BF01171116.
[19]	M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24 (1973), 229-238.
[20]	Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 253 (2013), 1774-1799. doi: 10.1016/j.jde.2012.11.008.
[21]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, preprint, arXiv:1301.6235.
[22]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Ivent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.
[23]	C. Li, A degree theory approach for the shooting method, preprint, arXiv:1301.6232.
[24]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.
[25]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270.
[26]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^N$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.
[27]	E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.
[28]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differ. Integral Equations, 9 (1996), 465-480.
[29]	Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon systems, Adv. Differential Equations, 17 (2012), 605-634.
[30]	Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differential Equations, 252 (2012), 2544-2562. doi: 10.1016/j.jde.2011.09.022.
[31]	P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8.
[32]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653.
[33]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.
[34]	E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
[35]	J. Villavert, Shooting with degree theory: Analysis of some weighted poly-harmonic systems, J. Differential Equations, 257 (2014), 1148-1167. doi: 10.1016/j.jde.2014.05.003.
[36]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.

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