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

Abstract

Let H n = C n × R $\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}$ be the n $n$ -dimensional Heisenberg group, Q = 2 n + 2 $Q=2n+2$ be the homogeneous dimension of H n $\mathbb {H}^{n}$ . In this paper, we investigate the existence of a least energy solution to the Q $Q$ -subLaplacian Schrödinger equation with either a constant V = γ $V=\gamma$ or a degenerate potential V $V$ vanishing on a bounded open subset of H n $\mathbb {H}^n$ : − div H ∇ H u Q − 2 ∇ H u + V ( ξ ) u Q − 2 u = f u $$\begin{equation} -\mathrm{div}_{\mathbb {H}}{\left({\left|\nabla _{\mathbb {H}}u\right|}^{Q-2} \nabla _{\mathbb {H}}u\right)} +V(\xi ) {\left|u\right|}^{Q-2}u=f{\left(u\right)} \end{equation}$$ (0.1)with the non-linear term f $f$ of maximal exponential growth exp ( α t Q Q − 1 ) $\exp (\alpha t^{\frac{Q}{Q-1}})$ as t → + ∞ $t\rightarrow +\infty$ . Since the Pólya–Szegö-type inequality fails on H n $\mathbb {H}^n$ , 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 H n $\mathbb {H}^n$ . 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 H n $\mathbb {H}^n$ . Second, we prove the existence of a least energy solution to the above equation with the constant potential V ( ξ ) = γ > 0 $V(\xi )=\gamma >0$ . Third, we establish the existence of a least energy solution to the Q $Q$ -subelliptic equation (0.1) involving the degenerate potential which vanishes on some open bounded set of H n $\mathbb {H}^{n}$ . We develop arguments that avoid using any symmetrization on H n $\mathbb {H}^n$ 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.