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January  2017, 37(1): 265-279. doi: 10.3934/dcds.2017011

Liouville theorems for stable solutions of the weighted Lane-Emden system

Hatem Hajlaoui 1,, , Abdellaziz Harrabi 2, and  Foued Mtiri 3,

1. 

Institut Préparatoire aux Etudes d'Ingénieurs, Université de Kairouan, Tunisie
2. 

Institut Supérieur des Mathématiques Appliquées et de l'Informatique, Université de Kairouan, Tunisie
3. 

Institut Élie Cartan de Lorraine, IECL, UMR 7502, Université de Lorraine, France

* Corresponding author: Hatem Hajlaoui

Received  February 2016 Revised  September 2016 Published  November 2016

We examine the general weighted Lane-Emden system
$-Δ u = ρ(x)v^p, -Δ v= ρ(x)u^θ, u,v>0 \;\mbox{in }\;\mathbb{R}^N$
where
$1 <p≤qθ$
 and
$ρ: \mathbb{R}^N \to \mathbb{R}$
 is a radial continuous function satisfying
$ρ(x)≥q A(1+|x|^2)^{\frac{α}{2}}$
 in
$\mathbb{R}^N$
 for some
$α≥q 0$
 and
$A>0$
. We prove some Liouville type results for stable solution and improve the previous works [2, 9, 12]. In particular, we establish a new comparison property (see Proposition 1 below) which is crucial to handle the case
$1 < p ≤q \frac{4}{3}$
. Our results can be applied also to the weighted Lane-Emden equation
$-Δ u = ρ(x)u^p$
 in
$\mathbb{R}^N$
.
Keywords: Stable solutions, Liouville type theorem, weighted Lane-Emden system.
Mathematics Subject Classification: Primary:35G30, 35B65;Secondary:35P3.
Citation: Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011
References:
[1]

W. Chen, L. Dupaigne and M. Ghergu, A new critical curve for the Lane-Emden system, Discrete Contin. Dyn. Syst., 34 (2014), 2469--2479.  doi: 10.3934/dcds.2014.34.2469.  Google Scholar

[2]

C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2013), 2357-2371.  doi: 10.1088/0951-7715/26/8/2357.  Google Scholar

[3]

C. Cowan, Regularity of stable solutions of a Lane-Emden type system, Methods Appl. Anal., 22 (2015), 301-311.  doi: 10.4310/MAA.2015.v22.n3.a4.  Google Scholar

[4]

C. Cowan and M. Fazly, On stable entire solutions of semilinear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.  doi: 10.1090/S0002-9939-2011-11351-0.  Google Scholar

[5]

C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. PDE., 49 (2014), 291-305.  doi: 10.1007/s00526-012-0582-4.  Google Scholar

[6]

J. Dávila, L. Dupaigne, K. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.  Google Scholar

[7]

L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solutions for the Liouville system, in: Geometric Partial Differential Equations, Publications of the Scuola Normale Superiore/CRM Series, 15 (2013), 139-144.  doi: 10.1007/978-88-7642-473-1_7.  Google Scholar

[8]

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

[9]

M. Fazly, Liouville type theorems for stable solutions of certain elliptic systems, Adv. Nonlinear Stud., 12 (2012), 1-17.  doi: 10.1515/ans-2012-0101.  Google Scholar

[10]

C. Gui, W. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\textbf{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar

[11]

H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of the biharmonic problem with polynomial growth, Pacific J. Math., 270 (2014), 79-93.  doi: 10.2140/pjm.2014.270.79.  Google Scholar

[12]

L. Hu, Liouville type results for semi-stable solutions of the weighted Lane-Emden system, J. Math. Anal. Appl., 432 (2015), 429-440.  doi: 10.1016/j.jmaa.2015.06.032.  Google Scholar

[13]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.   Google Scholar

[14]

E. Mitidieri and S. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.   Google Scholar

[15]

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

[16]

P. Polácik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part Ⅰ: Elliptic systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[17]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equations, 9 (1996), 635-653.   Google Scholar

[18]

P. 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

[19]

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]

W. Chen, L. Dupaigne and M. Ghergu, A new critical curve for the Lane-Emden system, Discrete Contin. Dyn. Syst., 34 (2014), 2469--2479.  doi: 10.3934/dcds.2014.34.2469.  Google Scholar

[2]

C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2013), 2357-2371.  doi: 10.1088/0951-7715/26/8/2357.  Google Scholar

[3]

C. Cowan, Regularity of stable solutions of a Lane-Emden type system, Methods Appl. Anal., 22 (2015), 301-311.  doi: 10.4310/MAA.2015.v22.n3.a4.  Google Scholar

[4]

C. Cowan and M. Fazly, On stable entire solutions of semilinear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.  doi: 10.1090/S0002-9939-2011-11351-0.  Google Scholar

[5]

C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. PDE., 49 (2014), 291-305.  doi: 10.1007/s00526-012-0582-4.  Google Scholar

[6]

J. Dávila, L. Dupaigne, K. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.  Google Scholar

[7]

L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solutions for the Liouville system, in: Geometric Partial Differential Equations, Publications of the Scuola Normale Superiore/CRM Series, 15 (2013), 139-144.  doi: 10.1007/978-88-7642-473-1_7.  Google Scholar

[8]

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

[9]

M. Fazly, Liouville type theorems for stable solutions of certain elliptic systems, Adv. Nonlinear Stud., 12 (2012), 1-17.  doi: 10.1515/ans-2012-0101.  Google Scholar

[10]

C. Gui, W. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\textbf{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar

[11]

H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of the biharmonic problem with polynomial growth, Pacific J. Math., 270 (2014), 79-93.  doi: 10.2140/pjm.2014.270.79.  Google Scholar

[12]

L. Hu, Liouville type results for semi-stable solutions of the weighted Lane-Emden system, J. Math. Anal. Appl., 432 (2015), 429-440.  doi: 10.1016/j.jmaa.2015.06.032.  Google Scholar

[13]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.   Google Scholar

[14]

E. Mitidieri and S. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.   Google Scholar

[15]

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

[16]

P. Polácik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part Ⅰ: Elliptic systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[17]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equations, 9 (1996), 635-653.   Google Scholar

[18]

P. 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

[19]

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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