Liouville type theorems for stable solutions of the weighted elliptic system

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

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