Liouville type theorems for stable solutions of the weighted elliptic system

Abstract

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