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

A new critical curve for the Lane-Emden system

Wenjing Chen 1, , Louis Dupaigne 2, and  Marius Ghergu 3,

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.
Keywords: singular solutions., stable solutions, critical curve, radially symmetric solutions, Lane-Emden system.
Mathematics Subject Classification: Primary: 35J47, 35J57; Secondary: 26D2.
Citation: Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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