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

Hatem Hajlaoui; Abdellaziz Harrabi; Foued Mtiri

doi:10.3934/dcds.2017011

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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

$ρ(x)≥q A(1+|x|^2)^{\frac{α}{2}}$

$\mathbb{R}^N$

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

$-Δ u = ρ(x)u^p$

$\mathbb{R}^N$

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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.
[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.
[15]	M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248.
[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.
[17]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equations, 9 (1996), 635-653.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.
[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.
[15]	M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248.
[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.
[17]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equations, 9 (1996), 635-653.
[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.
[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.

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