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 $1<p\le \theta$ and $\rho :{\mathbb{R}}^N\to \mathbb{R}$ is a radial continuous function satisfying $\rho (x)\ge A(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{\frac{\alpha }{2}}$ 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 $1 < p \leq \frac{4}{3}$. Our results can be applied also to the weighted Lane-Emden equation $-\mathrm{\Delta }u=\rho (x)u^p$ in ${\mathbb{R}}^N$.