Critical Quasilinear Schrödinger Equations on the Heisenberg Group: Existence and Nonexistence

Abstract

We study the quasilinear Schrödinger equation \begin{align*} -Δ_{\mathbb{H}} u +V(ξ)u-Δ_{\mathbb{H}} (\left|u\right|^{2α})\left|u\right|^{2α-2} u= λ\left|u\right|^{q-2}u + \left|u\right|^{p-2}u \quad \text{ in } \mathbb{H}^N, \end{align*} where ${\Delta }_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group ${\mathbb{H}}^N$, $4\alpha <q<p\le 2\alpha Q^{\ast }$, $\alpha >\frac{1}{2}$, and $2\alpha Q^{\ast }$ is the critical exponent, $Q=2N+2$ being the homogeneous dimension and $Q^{\ast }=\frac{2Q}{Q-2}$. For $p=2\alpha Q^{\ast }$ and $\lambda >0$, we obtain a nontrivial solution, assuming that the potential is bounded below by a positive constant and is either asymptotically constant from above or invariant under a discrete subgroup of ${\mathbb{H}}^N$. In the opposite direction, we prove a Pohožaev identity for ${\Delta }_{\mathbb{H}}$ and combine it with the Nehari identity, obtaining a family of identities from which the quasilinear energy disappears exactly at the exponent $2\alpha Q^{\ast }$. This yields a nonexistence theorem under a monotonicity condition on the potential with respect to anisotropic dilations and shows that no nontrivial solution exists for $\lambda \le 0$; the sign of the subcritical perturbation determines solvability. Along the way, we show that every weak solution is bounded and decays exponentially in the Korányi gauge.