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

We examine the general weighted Lane-Emden system \begin{align*} -\Delta u = \rho(x)v^p,\quad -\Delta v= \rho(x)u^\theta, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N \end{align*} where $1<p\leq\theta$ and $\rho: \mathbb{R}^N\rightarrow \mathbb{R}$ is a radial continuous function satisfying $\rho(x)\geq A(1+|x|^2)^{\frac{\alpha}{2}}$ in $\mathbb{R}^N$ for some $\alpha\geq 0$ 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 $1 < p \leq \frac{4}{3}$. Our results can be applied also to the weighted Lane-Emden equation $-\Delta u = \rho(x)u^p$ in $\mathbb{R}^N$.