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

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 & 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,, \emph{Indiana Univ. Math. J.}, 51 (2002), 37. Google Scholar
[2]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 42 (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar
[3]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, \emph{Compos. Math.}, 53 (1984), 259. 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,, \emph{Milan J. Math.}, 76 (2008), 27. doi: 10.1007/s00032-008-0090-3. Google Scholar
[5]	G. Caristi, E. Mitidieri and R. Soranzo, Isolated singularities of polyharmonic equations,, \emph{Atti Sem. Mat. Fis. Univ. Modena}, 46 (1998), 257. Google Scholar
[6]	F. Catrina and Z. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 229. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. Google Scholar
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[8]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar
[9]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2497. doi: 10.3934/cpaa.2013.12.2497. Google Scholar
[10]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. in Partial Differential Equations}, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar
[11]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 347. Google Scholar
[12]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar
[13]	L. D'Ambrosio and E. Mitidieri, Hardy-Littlewood-Sobolev systems and related Liouville theorems,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 7 (2014), 653. doi: 10.3934/dcdss.2014.7.653. Google Scholar
[14]	D. G. De Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems,, \emph{Ann. Sc. Norm. Sup. Pisa}, 21 (1994), 387. Google Scholar
[15]	B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Adv. Math. Suppl. Studies A}, 7 (1981), 369. Google Scholar
[16]	B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, \emph{Comm. Partial Differential Equations}, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar
[17]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure and Appl. Math.}, 34 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar
[18]	G. H. Hardy, J. E. Littlewood and G. Pólya, Some properties of fractional integral (1),, \emph{Math. Z.}, 27 (1928), 565. doi: 10.1007/BF01171116. Google Scholar
[19]	M. Hénon, Numerical experiments on the stability of spherical stellar systems,, \emph{Astronom. Astrophys.}, 24 (1973), 229. Google Scholar
[20]	Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations,, \emph{J. Differential Equations}, 253 (2013), 1774. doi: 10.1016/j.jde.2012.11.008. Google Scholar
[21]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, preprint, (). Google Scholar
[22]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, \emph{Ivent. Math.}, 123 (1996), 221. doi: 10.1007/s002220050023. Google Scholar
[23]	C. Li, A degree theory approach for the shooting method,, preprint, (). Google Scholar
[24]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349. doi: 10.2307/2007032. Google Scholar
[25]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{J. Partial Differential Equations}, 19 (2006), 256. Google Scholar
[26]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^N$,, \emph{J. Differential Equations}, 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016. Google Scholar
[27]	E. Mitidieri, A Rellich type identity and applications,, \emph{Comm. Partial Differential Equations}, 18 (1993), 125. doi: 10.1080/03605309308820923. Google Scholar
[28]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465. Google Scholar
[29]	Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon systems,, \emph{Adv. Differential Equations}, 17 (2012), 605. Google Scholar
[30]	Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations,, \emph{J. Differential Equations}, 252 (2012), 2544. doi: 10.1016/j.jde.2011.09.022. Google Scholar
[31]	P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8. Google Scholar
[32]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635. Google Scholar
[33]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Adv. Math.}, 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014. Google Scholar
[34]	E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, \emph{J. Math. Mech.}, 7 (1958), 503. Google Scholar
[35]	J. Villavert, Shooting with degree theory: Analysis of some weighted poly-harmonic systems,, \emph{J. Differential Equations}, 257 (2014), 1148. doi: 10.1016/j.jde.2014.05.003. Google Scholar
[36]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

show all references

References:

[1]	J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems,, \emph{Indiana Univ. Math. J.}, 51 (2002), 37. Google Scholar
[2]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 42 (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar
[3]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, \emph{Compos. Math.}, 53 (1984), 259. 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,, \emph{Milan J. Math.}, 76 (2008), 27. doi: 10.1007/s00032-008-0090-3. Google Scholar
[5]	G. Caristi, E. Mitidieri and R. Soranzo, Isolated singularities of polyharmonic equations,, \emph{Atti Sem. Mat. Fis. Univ. Modena}, 46 (1998), 257. Google Scholar
[6]	F. Catrina and Z. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 229. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. Google Scholar
[7]	W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems on a half space,, \emph{J. Funct. Anal.}, 265 (2013), 1522. doi: 10.1016/j.jfa.2013.06.010. Google Scholar
[8]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar
[9]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2497. doi: 10.3934/cpaa.2013.12.2497. Google Scholar
[10]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. in Partial Differential Equations}, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar
[11]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 347. Google Scholar
[12]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar
[13]	L. D'Ambrosio and E. Mitidieri, Hardy-Littlewood-Sobolev systems and related Liouville theorems,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 7 (2014), 653. doi: 10.3934/dcdss.2014.7.653. Google Scholar
[14]	D. G. De Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems,, \emph{Ann. Sc. Norm. Sup. Pisa}, 21 (1994), 387. Google Scholar
[15]	B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Adv. Math. Suppl. Studies A}, 7 (1981), 369. Google Scholar
[16]	B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, \emph{Comm. Partial Differential Equations}, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar
[17]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure and Appl. Math.}, 34 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar
[18]	G. H. Hardy, J. E. Littlewood and G. Pólya, Some properties of fractional integral (1),, \emph{Math. Z.}, 27 (1928), 565. doi: 10.1007/BF01171116. Google Scholar
[19]	M. Hénon, Numerical experiments on the stability of spherical stellar systems,, \emph{Astronom. Astrophys.}, 24 (1973), 229. Google Scholar
[20]	Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations,, \emph{J. Differential Equations}, 253 (2013), 1774. doi: 10.1016/j.jde.2012.11.008. Google Scholar
[21]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, preprint, (). Google Scholar
[22]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, \emph{Ivent. Math.}, 123 (1996), 221. doi: 10.1007/s002220050023. Google Scholar
[23]	C. Li, A degree theory approach for the shooting method,, preprint, (). Google Scholar
[24]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349. doi: 10.2307/2007032. Google Scholar
[25]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{J. Partial Differential Equations}, 19 (2006), 256. Google Scholar
[26]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $R^N$,, \emph{J. Differential Equations}, 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016. Google Scholar
[27]	E. Mitidieri, A Rellich type identity and applications,, \emph{Comm. Partial Differential Equations}, 18 (1993), 125. doi: 10.1080/03605309308820923. Google Scholar
[28]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465. Google Scholar
[29]	Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon systems,, \emph{Adv. Differential Equations}, 17 (2012), 605. Google Scholar
[30]	Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations,, \emph{J. Differential Equations}, 252 (2012), 2544. doi: 10.1016/j.jde.2011.09.022. Google Scholar
[31]	P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8. Google Scholar
[32]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635. Google Scholar
[33]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Adv. Math.}, 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014. Google Scholar
[34]	E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, \emph{J. Math. Mech.}, 7 (1958), 503. Google Scholar
[35]	J. Villavert, Shooting with degree theory: Analysis of some weighted poly-harmonic systems,, \emph{J. Differential Equations}, 257 (2014), 1148. doi: 10.1016/j.jde.2014.05.003. Google Scholar
[36]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

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