Liouville type theorems for stable solutions of the weighted elliptic system

We examine the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=(1+|x|^2)^{\frac{\alpha}{2}} v,\ -\Delta v=(1+|x|^2)^{\frac{\alpha}{2}} u^p, \end{cases} \quad \mbox{in};
$\mathbb{R}$^N, \end{equation*}where $N\ge 5$, $p>1$ and $\alpha >0$. We prove Liouville type results for the classical positive (nonnegative) stable solutions in dimension $N<\mathrm{\ell }+{\displaystyle \frac{\alpha (\mathrm{\ell }-2)}{2}}$ ($N <\ell+\dfrac{\alpha (\ell-2)(p+3)}{4(p+1)}$)and$\ell \ge 5$,$p \in (1,p_*(\ell))$\mathrm{.}Inparticular,forany$p>1$and$\alpha > 0$, we obtain the nonexistence of classical positive (nonnegative) stable solutions for any $N \le 12+5 \alpha$($N\le 12+\dfrac{5\alpha (p+3)}{2(p+1)}$).