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Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians
A Liouville theorem for the subcritical Lane-Emden system
1. | Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA |
2. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
The Lane-Emden conjecture says that the subcritical Lane-Emden system admits no positive solution. In this paper, we present a necessary and sufficient condition to the Lane-Emden conjecture. This condition is an energy-type a priori estimate. The necessity of the condition we found can be easily checked. However, a major difficulty lies in the sufficiency. The proof is quite involving, but the benefit is that it reduces the longstanding problem to obtaining the a priori estimate of energy type.
References:
[1] |
I. Birindelli and E. Mitidieri,
Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247.
doi: 10.1017/S0308210500027293. |
[2] |
J. Busca and R. Manasevich,
A Liouville-type theorem for Lane-Emden systems, Indiana University Mathematics Journal, 51 (2002), 37-51.
|
[3] |
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. |
[4] |
W. Chen and C. Li,
An integral system and the Lane-Emden conjecture, Disc. & Cont. Dynamics Sys., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[5] |
Ph. Clément, D. G. de Figueiredo and E. Mitidieri,
Positive solutions of semilinear elliptic systems, Communications in Partial Differential Equations, 17 (1992), 923-940.
doi: 10.1080/03605309208820869. |
[6] |
D. De Figueiredo and P. Felmer,
A liouville-type theorem for elliptic systems, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 21 (1994), 387-397.
|
[7] |
B. Gidas and J. Spruck,
Global and local behavior of positive solutions of nonlinear elliptic equations, Communications on Pure and Applied Mathematics, 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[8] |
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. |
[9] |
C. Li, A degree theory approach for the shooting method, arXiv preprint, arXiv:1301.6232, 2013. |
[10] |
C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, https://doi.org/10.1073/pnas.1804225115, 2018. |
[11] |
J. Liu, Y. Guo and Y. Zhang,
Existence of positive entire solutions for polyharmonic equations and systems, Journal of Partial Differential Equations, 19 (2006), 256-270.
|
[12] |
E. Mitidieri,
A Rellich type identity and applications: Identity and applications, Communications in Partial Differential Equations, 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[13] |
E. Mitidieri,
Nonexistence of positive solutions of semilinear elliptic systems in RN, Differ. Integral Equations, 9 (1996), 465-479.
|
[14] |
Q. Phan,
Liouville-type theorems and bounds of solutions for hardy-hénon elliptic systems, Advances in Differential Equations, 17 (2012), 605-634.
|
[15] |
Q. Phan and P. Souplet,
Liouville-type theorems and bounds of solutions of hardy-hénon equations, Journal of Differential Equations, 252 (2012), 2544-2562.
doi: 10.1016/j.jde.2011.09.022. |
[16] |
P. Polacik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I : Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[17] |
P. Quittner and P. Souplet,
Optimal Liouville-type theorems for noncooperative elliptic schrödinger systems and applications, Communications in Mathematical Physics, 311 (2012), 1-19.
doi: 10.1007/s00220-012-1440-0. |
[18] |
W. Reichel and H. Zou,
Non-existence results for semilinear cooperative elliptic systems via moving spheres, Journal of Differential Equations, 161 (2000), 219-243.
doi: 10.1006/jdeq.1999.3700. |
[19] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653.
|
[20] |
J. Serrin and H. Zou,
Existence of positive solutions of the Lane-Emden system, Atti Semi. Mat. Fis. Univ. Modena, 46 (1998), 369-380.
|
[21] |
P. Souplet,
The proof of the Lane-Emden conjecture in four space dimensions, Advances in Mathematics, 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[22] |
P. Souplet,
Liouville-type theorems for elliptic schrödinger systems associated with copositive matrices, Networks & Heterogeneous Media, 7 (2012), 967-988.
doi: 10.3934/nhm.2012.7.967. |
[23] |
J. Villavert,
A refined approach for non-negative entire solutions of ∆u + up = 0 with subcritical sobolev growth, Adv. Nonlinear Stud., 17 (2017), 691-703.
doi: 10.1515/ans-2016-6024. |
show all references
References:
[1] |
I. Birindelli and E. Mitidieri,
Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247.
doi: 10.1017/S0308210500027293. |
[2] |
J. Busca and R. Manasevich,
A Liouville-type theorem for Lane-Emden systems, Indiana University Mathematics Journal, 51 (2002), 37-51.
|
[3] |
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. |
[4] |
W. Chen and C. Li,
An integral system and the Lane-Emden conjecture, Disc. & Cont. Dynamics Sys., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[5] |
Ph. Clément, D. G. de Figueiredo and E. Mitidieri,
Positive solutions of semilinear elliptic systems, Communications in Partial Differential Equations, 17 (1992), 923-940.
doi: 10.1080/03605309208820869. |
[6] |
D. De Figueiredo and P. Felmer,
A liouville-type theorem for elliptic systems, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 21 (1994), 387-397.
|
[7] |
B. Gidas and J. Spruck,
Global and local behavior of positive solutions of nonlinear elliptic equations, Communications on Pure and Applied Mathematics, 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[8] |
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. |
[9] |
C. Li, A degree theory approach for the shooting method, arXiv preprint, arXiv:1301.6232, 2013. |
[10] |
C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, https://doi.org/10.1073/pnas.1804225115, 2018. |
[11] |
J. Liu, Y. Guo and Y. Zhang,
Existence of positive entire solutions for polyharmonic equations and systems, Journal of Partial Differential Equations, 19 (2006), 256-270.
|
[12] |
E. Mitidieri,
A Rellich type identity and applications: Identity and applications, Communications in Partial Differential Equations, 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[13] |
E. Mitidieri,
Nonexistence of positive solutions of semilinear elliptic systems in RN, Differ. Integral Equations, 9 (1996), 465-479.
|
[14] |
Q. Phan,
Liouville-type theorems and bounds of solutions for hardy-hénon elliptic systems, Advances in Differential Equations, 17 (2012), 605-634.
|
[15] |
Q. Phan and P. Souplet,
Liouville-type theorems and bounds of solutions of hardy-hénon equations, Journal of Differential Equations, 252 (2012), 2544-2562.
doi: 10.1016/j.jde.2011.09.022. |
[16] |
P. Polacik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I : Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[17] |
P. Quittner and P. Souplet,
Optimal Liouville-type theorems for noncooperative elliptic schrödinger systems and applications, Communications in Mathematical Physics, 311 (2012), 1-19.
doi: 10.1007/s00220-012-1440-0. |
[18] |
W. Reichel and H. Zou,
Non-existence results for semilinear cooperative elliptic systems via moving spheres, Journal of Differential Equations, 161 (2000), 219-243.
doi: 10.1006/jdeq.1999.3700. |
[19] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653.
|
[20] |
J. Serrin and H. Zou,
Existence of positive solutions of the Lane-Emden system, Atti Semi. Mat. Fis. Univ. Modena, 46 (1998), 369-380.
|
[21] |
P. Souplet,
The proof of the Lane-Emden conjecture in four space dimensions, Advances in Mathematics, 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[22] |
P. Souplet,
Liouville-type theorems for elliptic schrödinger systems associated with copositive matrices, Networks & Heterogeneous Media, 7 (2012), 967-988.
doi: 10.3934/nhm.2012.7.967. |
[23] |
J. Villavert,
A refined approach for non-negative entire solutions of ∆u + up = 0 with subcritical sobolev growth, Adv. Nonlinear Stud., 17 (2017), 691-703.
doi: 10.1515/ans-2016-6024. |
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