2022/11/10 by Lu Chen, Guozhen Lu, Maochun Zhu · 2 citations
Mathematics · Computer Science · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering
paper · doi:10.1112/plms.12495
Let H n = C n × R \mathbb Hn=\mathbb Cn× \mathbb R be the n n -dimensional Heisenberg group, Q = 2 n + 2 Q=2n+2 be the homogeneous dimension of H n \mathbb Hn . 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=γ or a degenerate potential V V vanishing on a bounded open subset of H n \mathbb Hn : − div H ∇ H u Q − 2 ∇ H u + V ( ξ ) u Q − 2 u = f u \beginequation -div_\mathbb H(|∇ _\mathbb Hu|Q-2 ∇ _\mathbb Hu) +V(ξ ) |u|Q-2u=f(u) \endequation (0.1)with the non-linear term f f of maximal exponential growth exp ( α t Q Q − 1 ) exp (α t(Q)/(Q-1)) as t → + ∞ t→ +∞ . Since the Pólya–Szegö-type inequality fails on H n \mathbb Hn , 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 Hn . 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 Hn . Second, we prove the existence of a least energy solution to the above equation with the constant potential V ( ξ ) = γ > 0 V(ξ )=γ >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 Hn . We develop arguments that avoid using any symmetrization on H n \mathbb Hn 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.