Least energy solutions to quasilinear subelliptic equations with constant and degenerate potentials on the Heisenberg group

Let (Formula presented.) be the (Formula presented.) -dimensional Heisenberg group, (Formula presented.) be the homogeneous dimension of (Formula presented.). In this paper, we investigate the existence of a least energy solution to the (Formula presented.) -subLaplacian Schrödinger equation with either a constant (Formula presented.) or a degenerate potential (Formula presented.) vanishing on a bounded open subset of (Formula presented.) : 0.1 (Formula presented.) with the non-linear term (Formula presented.) of maximal exponential growth (Formula presented.) as (Formula presented.). Since the Pólya–Szegö-type inequality fails on (Formula presented.), the coercivity of the potential has been a standard assumption in the literature for subelliptic equations to exclude the vanishing phenomena of Palais–Smale sequence on the entire space (Formula presented.). Our aim in this paper is to remove this strong assumption. To this end, we first establish a sharp critical Trudinger–Moser inequality involving a degenerate potential on (Formula presented.). Second, we prove the existence of a least energy solution to the above equation with the constant potential (Formula presented.). Third, we establish the existence of a least energy solution to the (Formula presented.) -subelliptic equation (0.1) involving the degenerate potential which vanishes on some open bounded set of (Formula presented.). We develop arguments that avoid using any symmetrization on (Formula presented.) where the Pólya–Szegö inequality fails. Fourth, we also establish the existence of a least energy solution to (0.1) when the potential is a non-degenerate Rabinowitz type potential but still fails to be coercive. Our results in this paper improve significantly on the earlier ones on quasilinear Schrödinger equations on the Heisenberg group in the literature. We note that all the main results and their proofs in this paper hold on stratified groups with the same proofs.