A new critical curve for the Lane-Emden system

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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

Abstract
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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 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.

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