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 $N<\ell+\dfrac{\alpha (\ell-2)}{2}$ ($N <\ell+\dfrac{\alpha (\ell-2)(p+3)}{4(p+1)}$) and $\ell \ge 5$, $p \in (1,p_*(\ell))$. In particular, for any $p>1$ and $\alpha > 0$, we obtain the nonexistence of classical positive (nonnegative) stable solutions for any $N \le 12+5 \alpha$ ($N\le 12+\dfrac{5\alpha (p+3)}{2(p+1)}$).