We study the Liouville-type theorems for the sub-elliptic inequality $-{\mathrm{\Delta }}_{\lambda }u\ge u^p\mathrm{i}\mathrm{n}{\mathbb{R}}^N$ and for the corresponding system $\begin{array}{l}-{\mathrm{\Delta }}_{\lambda }u\ge v^p\\ -{\mathrm{\Delta }}_{\lambda }v\ge u^q\end{array}\mathrm{i}\mathrm{n}{\mathbb{R}}^N,$ where $p,q\in \mathbb{R}$, and ${\mathrm{\Delta }}_{\lambda }$ is a strongly degenerate operator of the form ${\mathrm{\Delta }}_{\lambda }=\sum _{i=1}^N{\mathrm{\partial }}_{x_i}\left({\lambda }_i^2{\mathrm{\partial }}_{x_i}\right).$ Here the functions ${\lambda }_i,i=1,2,\dots ,N,$ satisfy certain conditions such that ${\mathrm{\Delta }}_{\lambda }$ is homogeneous of degree 2 with respect to a group of dilations in ${\mathbb{R}}^N$.

We first prove that the scalar inequality has no positive solution provided $-\mathrm{\infty }<p\le \frac{Q}{Q-2},$ where Q is the homogeneous dimension of ${\mathbb{R}}^N$ associated to the operator ${\mathrm{\Delta }}_{\lambda }$.

Then, we establish the nonexistence of positive solutions of the corresponding system in one of following cases:

1. $p\le 0$ or $q\le 0$,

2. p, q>0 and $pq\le 1$,

3. p, q>0, pq>1 and $max\left\{\frac{2(p+1)}{pq-1},\frac{2(q+1)}{pq-1}\right\}>Q-2$.

Our proofs are based on a maximum principle argument combined with a refinement of Souto's reduction.

我们研究了亚椭圆不等式的 Liouville 型定理 $-{\mathrm{\Delta }}_{\lambda }你\ge {你}^p\mathrm{一世}\mathrm{n}{\mathbb{电阻}}^N$ 以及对应的系统 $\begin{array}{l}-{\mathrm{\Delta }}_{\lambda }你\ge v^p\\ -{\mathrm{\Delta }}_{\lambda }v\ge {你}^q\end{array}\mathrm{一世}\mathrm{n}{\mathbb{电阻}}^N,$ 在哪里 $p,q\in \mathbb{电阻}$， 和 ${\mathrm{\Delta }}_{\lambda }$ 是形式的强退化运算符 ${\mathrm{\Delta }}_{\lambda }=\sum _{\mathrm{一世}=1}^N{\mathrm{\partial }}_{X_{\mathrm{一世}}}\left({\lambda }_{\mathrm{一世}}^2{\mathrm{\partial }}_{X_{\mathrm{一世}}}\right).$ 这里的功能 ${\lambda }_{\mathrm{一世}},\mathrm{一世}=1,2,\dots ,N,$ 满足某些条件使得 ${\mathrm{\Delta }}_{\lambda }$ 相对于一组膨胀是 2 次齐次的 ${\mathbb{电阻}}^N$.

我们首先证明标量不等式没有正解提供 $-\mathrm{\infty }<p\le \frac{问}{问-2},$其中Q是齐次维数${\mathbb{电阻}}^N$ 与运营商相关联 ${\mathrm{\Delta }}_{\lambda }$.

然后，我们在以下情况之一中建立相应系统的正解不存在：

1. $p\le 0$ 或者 $q\le 0$,

2. p , q > 0 和$pq\le 1$,

3. p , q >0, pq >1 和$最大限度\left\{\frac{2(p+1)}{pq-1},\frac{2(q+1)}{pq-1}\right\}>问-2$.

我们的证明基于最大原理论证并结合了 Souto 约简的改进。