In this paper, we study the following sub-elliptic systems of inequalities \(\begin{aligned} \left\{ \begin{array}{lcc} \Delta _\mathbb {H}u+h_1(\xi )v^p\le 0\ \ \text{ in }\ \Omega ,\\ \Delta _\mathbb {H}v+h_2(\xi )u^q\le 0\ \ \text{ in }\ \Omega , \end{array} \right. \ \text{ and }\ \ \ \left\{ \begin{array}{lcc} \Delta _\mathbb {H}u+h_1(\xi )v^p\le 0\ \ \text{ in }\ \Omega ,\\ \Delta _\mathbb {H}v+h_2(\xi )w^q\le 0\ \ \text{ in }\ \Omega ,\\ \Delta _\mathbb {H}w+h_3(\xi )u^s\le 0\ \ \text{ in }\ \Omega , \end{array} \right. \end{aligned}\) where \(\Delta _\mathbb {H}\) denotes the Heisenberg Laplacian in the Heisenberg group \(\mathbb {H}^n(n\ge 1)\) , and \(h_i\) \((i=1,2,3)\) are some non-negative functions, and \(\Omega \subset \mathbb {H}^n\) is an unbounded domain. By using the test functions method and some analysis techniques, we prove that, under suitable conditions on \(h_i\) , p, q, s and \(\Omega \) , the above sub-elliptic systems do not possess positive solutions.
