Liouville-type theorems for sub-elliptic systems involving Δ_λ-Laplacian

We study the Liouville-type theorems for the sub-elliptic inequality  and for the corresponding system  ,  in , where  and  is a strongly degenerate operator of the form . Here the functions , satisfy certain conditions such that  is homogeneous of degree 2 with respect to a group of dilations in .

We first prove that the scalar inequality has no positive solution provided where Q is the homogeneous dimension of  associated to the operator .

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

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