Critical Quasilinear Schr\"odinger Equations on the Heisenberg Group: Existence and Nonexistence

Abstract

We study the quasilinear Schr\"odinger equation \begin{align*} -\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \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 $\mathbb{H}^N$, $4\alpha \frac12$, and $2\alpha Q^{*}$ is the critical exponent, $Q=2N+2$ being the homogeneous dimension and $Q^{*}=\frac{2Q}{Q-2}$. For $p=2\alpha Q^{*}$ 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 Poho\v{z}aev 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 $2\alpha Q^{*}$. 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 $\lambda \leq 0$; the sign of the subcritical perturbation determines solvability. Along the way, we show that every weak solution is bounded and decays exponentially in the Kor\'anyi gauge.