Critical Kirchhoff equations involving the -sub-Laplacians operators on the Heisenberg group

In this paper, we deal with a class of Kirchhoff-type critical elliptic equations involving the [Formula: see text]-sub-Laplacians operators on the Heisenberg group of the form M(∥DHu∥pp + ∥u∥ p,Vp)[−Δ H,pu + V (ξ)|u|p−2u] = λf(ξ,u) + |u|p∗−2u,ξ ∈ ℍn,u ∈HWV1,p(ℍn), where [Formula: see text] is the [Formula: see text]-sub-Laplacian, [Formula: see text], [Formula: see text] is the horizontal Sobolev space on [Formula: see text]. And [Formula: see text] is the homogeneous dimension of [Formula: see text], [Formula: see text] is a real parameter, [Formula: see text] is the critical Sobolev exponent on the Heisenberg group. Under some proper assumptions on the Kirchhoff function [Formula: see text], the potential function [Formula: see text] and [Formula: see text], together with the mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases on the Heisenberg group. The results of this paper extend or complete recent papers and are new in several directions for the non-degenerate and degenerate critical Kirchhoff equations involving the [Formula: see text]-Laplacian type operators on the Heisenberg group.