# American Institute of Mathematical Sciences

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.

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##### 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.
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