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

Abstract

We examine the general weighted Lane-Emden system \begin{align*} -Δu = ρ(x)v^p,\quad -Δv= ρ(x)u^θ, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N \end{align*} where $1<p\le \theta$ and $\rho :{\mathbb{R}}^N\to \mathbb{R}$ is a radial continuous function satisfying $\rho (x)\ge A(1+|x{|}^2{)}^{\frac{\alpha }{2}}$ in ${\mathbb{R}}^N$ for some $\alpha \ge 0$ and $A>0$. We prove some Liouville type results for stable solution and improve the previous works \cite{co, Fa, HU}. In particular, we establish a new comparison property (see Proposition 1.1 below) which is crucial to handle the case $1<p\le \frac{4}{3}$. Our results can be applied also to the weighted Lane-Emden equation $-\Delta u=\rho (x)u^p$ in ${\mathbb{R}}^N$.