# American Institute of Mathematical Sciences

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

## 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

* 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.

Citation: Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete and 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. [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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##### 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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