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March  2019, 39(3): 1359-1377. doi: 10.3934/dcds.2019058

A Liouville theorem for the subcritical Lane-Emden system

Ze Cheng 1, and  Genggeng Huang 2,,

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

* Corresponding author: Genggeng Huang

Received  May 2018 Published  December 2018

Fund Project: The first author is partially supported by NSF DMS-1405175. The second author is partially supported by NSFC-11401376 and the Scholarship of International Postdoctoral Exchange Fellowship Program.

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.

Keywords: Lane-Emden conjecture, energy a priori estimates.
Mathematics Subject Classification: Primary: 35B09, 35B53, 35J05, 35J61.
Citation: Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058
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.  Google Scholar

[2]

J. Busca and R. Manasevich, A Liouville-type theorem for Lane-Emden systems, Indiana University Mathematics Journal, 51 (2002), 37-51.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Li, A degree theory approach for the shooting method, arXiv preprint, arXiv:1301.6232, 2013. Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in RN, Differ. Integral Equations, 9 (1996), 465-479.   Google Scholar

[14]

Q. Phan, Liouville-type theorems and bounds of solutions for hardy-hénon elliptic systems, Advances in Differential Equations, 17 (2012), 605-634.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Busca and R. Manasevich, A Liouville-type theorem for Lane-Emden systems, Indiana University Mathematics Journal, 51 (2002), 37-51.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Li, A degree theory approach for the shooting method, arXiv preprint, arXiv:1301.6232, 2013. Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in RN, Differ. Integral Equations, 9 (1996), 465-479.   Google Scholar

[14]

Q. Phan, Liouville-type theorems and bounds of solutions for hardy-hénon elliptic systems, Advances in Differential Equations, 17 (2012), 605-634.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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