Liouville-type theorems for an elliptic system involving fractional Laplacian operators with mixed order

We study the nonexistence of nontrivial solutions for the nonlinear elliptic system $$\displaylines{ G_{\alpha,\beta,\theta}(u^{p},u^{q}) = v^{r}\cr G_{\lambda,\mu,\theta}(v^{s},v^{t}) = u^{m}\cr u,v\geq 0, }$$ where $0<\alpha,\beta,\lambda,\mu\leq 2$, $\theta\geq 0$, $m>q\geq p\geq 1$, $r>t\geq s\geq 1$, and $G_{\alpha,\beta,\theta}$ is the fractional operator of mixed orders $\alpha,\beta$, defined by $$ G_{\alpha,\beta,\theta}(u,v)=(-\Delta_x)^{\alpha/2}u +|x|^{2\theta} (-\Delta_y)^{\beta/2}v, \quad \text{in }\mathbb{R}^{N_1} \times \mathbb{R}^{N_2}. $$ Here, $(-\Delta_x)^{\alpha/2}$, $0<\alpha<2$, is the fractional Laplacian operator of order $\alpha/2$ with respect to the variable $x\in \mathbb{R}^{N_1}$, and $(-\Delta_y)^{\beta/2}$, $0<\beta