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

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 $1<p\leqθ$ and $ρ: \mathbb{R}^N\rightarrow \mathbb{R}$ is a radial continuous function satisfying $ρ(x)\geq A(1+|x|^2)^{\fracα{2}}$ in $\mathbb{R}^N$ for some $α\geq 0$ 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 $-Δu = ρ(x)u^p$ in $\mathbb{R}^N$.

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