June  2014, 34(6): 2469-2479. doi: 10.3934/dcds.2014.34.2469

A new critical curve for the Lane-Emden system

1. 

Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

2. 

Institut Camille Jordan UMR CNRS 5208, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

3. 

School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Received  February 2013 Revised  May 2013 Published  December 2013

We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\mathbb{R}^N$, $-\Delta v=u^q$ in $\mathbb{R}^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.
Citation: Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2469-2479. doi: 10.3934/dcds.2014.34.2469
References:
[1]

J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 543-590. doi: 10.1016/j.anihpc.2003.06.001.

[2]

P. Caldiroli and R. Musina, Rellich inequalities with weights, Calc. Var. Partial Differential Equations, 45 (2012), 147-164. doi: 10.1007/s00526-011-0454-3.

[3]

C. Cowan, Regularity of stable solutions of a Lane-Emden type system,, preprint, (). 

[4]

C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2003), 2357-2371. doi: 10.1088/0951-7715/26/8/2357.

[5]

J. Dávila, L. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232. doi: 10.1016/j.jfa.2010.12.028.

[6]

J. Dávila, L. Dupaigne and M. Montenegro, The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal., 7 (2008), 795-817. doi: 10.3934/cpaa.2008.7.795.

[7]

S. Dumont, L. Dupaigne, O. Goubet and V. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.

[8]

L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault, The Gel'fand problem for the biharmonic operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752. doi: 10.1007/s00205-013-0613-0.

[9]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbbR^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001.

[10]

H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of biharmonic problem with polynomial growth,, preprint, (). 

[11]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241. 

[12]

F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936. doi: 10.1007/s00208-005-0748-x.

[13]

P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661. doi: 10.1088/0951-7715/22/7/009.

[14]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.

[15]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differential Integral Equations, 9 (1996), 465-479.

[16]

M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248.

[17]

P. Poláčik, P. Quittner and Ph. 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.

[18]

J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, in A tribute to Ilya Bakelman (eds. I. R. Bakelman, S. A. Fulling and S. D. Taliaferro) (College Station, TX, 1993), Discourses Math. Appl., 3, Texas A & M Univ., College Station, TX, 1994, 55-68.

[19]

Ph. 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.

[20]

G. Sweers, Strong positivity in $C(\overline\Omega)$ for elliptic systems, Math. Z., 209 (1992), 251-271. doi: 10.1007/BF02570833.

[21]

R. C. A. M Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1992), 375-398. doi: 10.1007/BF00375674.

[22]

J. Wei, X. Xu and Y. Wen, On the classification of stable solutions to biharmonic problems in large dimensions, Pacific J. Math., 263 (2013), 495-512. doi: 10.2140/pjm.2013.263.495.

[23]

J. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612. doi: 10.1007/s00208-012-0894-x.

show all references

References:
[1]

J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 543-590. doi: 10.1016/j.anihpc.2003.06.001.

[2]

P. Caldiroli and R. Musina, Rellich inequalities with weights, Calc. Var. Partial Differential Equations, 45 (2012), 147-164. doi: 10.1007/s00526-011-0454-3.

[3]

C. Cowan, Regularity of stable solutions of a Lane-Emden type system,, preprint, (). 

[4]

C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2003), 2357-2371. doi: 10.1088/0951-7715/26/8/2357.

[5]

J. Dávila, L. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232. doi: 10.1016/j.jfa.2010.12.028.

[6]

J. Dávila, L. Dupaigne and M. Montenegro, The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal., 7 (2008), 795-817. doi: 10.3934/cpaa.2008.7.795.

[7]

S. Dumont, L. Dupaigne, O. Goubet and V. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.

[8]

L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault, The Gel'fand problem for the biharmonic operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752. doi: 10.1007/s00205-013-0613-0.

[9]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbbR^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001.

[10]

H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of biharmonic problem with polynomial growth,, preprint, (). 

[11]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241. 

[12]

F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936. doi: 10.1007/s00208-005-0748-x.

[13]

P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661. doi: 10.1088/0951-7715/22/7/009.

[14]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.

[15]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differential Integral Equations, 9 (1996), 465-479.

[16]

M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248.

[17]

P. Poláčik, P. Quittner and Ph. 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.

[18]

J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, in A tribute to Ilya Bakelman (eds. I. R. Bakelman, S. A. Fulling and S. D. Taliaferro) (College Station, TX, 1993), Discourses Math. Appl., 3, Texas A & M Univ., College Station, TX, 1994, 55-68.

[19]

Ph. 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.

[20]

G. Sweers, Strong positivity in $C(\overline\Omega)$ for elliptic systems, Math. Z., 209 (1992), 251-271. doi: 10.1007/BF02570833.

[21]

R. C. A. M Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1992), 375-398. doi: 10.1007/BF00375674.

[22]

J. Wei, X. Xu and Y. Wen, On the classification of stable solutions to biharmonic problems in large dimensions, Pacific J. Math., 263 (2013), 495-512. doi: 10.2140/pjm.2013.263.495.

[23]

J. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612. doi: 10.1007/s00208-012-0894-x.

[1]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[2]

Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011

[3]

Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1681-1698. doi: 10.3934/cpaa.2021036

[4]

Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094

[5]

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

[6]

Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167

[7]

Julián López-Gómez. Uniqueness of radially symmetric large solutions. Conference Publications, 2007, 2007 (Special) : 677-686. doi: 10.3934/proc.2007.2007.677

[8]

Philip Korman, Junping Shi. On lane-emden type systems. Conference Publications, 2005, 2005 (Special) : 510-517. doi: 10.3934/proc.2005.2005.510

[9]

Lu Chen, Guozhen Lu, Yansheng Shen. Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on $ \mathbb{S}^n $. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022073

[10]

Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 793-805. doi: 10.3934/dcdss.2014.7.793

[11]

Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427

[12]

J. Ignacio Tello. Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022045

[13]

Yuxia Guo, Ting Liu. Liouville-type theorem for high order degenerate Lane-Emden system. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2073-2100. doi: 10.3934/dcds.2021184

[14]

Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297

[15]

Harunori Monobe. Behavior of radially symmetric solutions for a free boundary problem related to cell motility. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 989-997. doi: 10.3934/dcdss.2015.8.989

[16]

István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003

[17]

Jianqing Chen, Qian Zhang. Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term. Communications on Pure and Applied Analysis, 2022, 21 (2) : 669-686. doi: 10.3934/cpaa.2021193

[18]

Jingbo Dou, Qianqiao Guo. Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains. Communications on Pure and Applied Analysis, 2012, 11 (2) : 453-464. doi: 10.3934/cpaa.2012.11.453

[19]

Linlin Dou. Singular solutions of Toda system in high dimensions. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3119-3142. doi: 10.3934/dcds.2022011

[20]

Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (116)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]