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Sharp existence criteria for positive solutions of Hardy--Sobolev type systems
1. | Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, China |
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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