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

Hatem Hajlaoui; Abdellaziz Harrabi; Foued Mtiri

doi:10.3934/dcds.2017011

Article Contents

2017, Volume 37, Issue 1: 265-279. Doi: 10.3934/dcds.2017011

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

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
* Corresponding author: Hatem Hajlaoui

Received: February 2016

Revised: September 2016

Published: September 2017

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Abstract

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 $\mathbb{R}^N$.

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Mathematics Subject Classification: Primary:35G30, 35B65;Secondary:35P30.

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