Modified $Q$-Laplacian Problem with Parameter and Exponential Nonlinearity on the Heisenberg Group

In this article, we investigate the following modified quasilinear equation driven by the $Q$-Laplacian: $$\begin{cases} -\Delta_Q u - \Delta_Q\bigl(|u|^{2\alpha}\bigr)\,|u|^{2\alpha-2} u = \lambda f(\xi,u) & \text{in } \Omega, \\[2mm] u = 0 & \text{on } \partial\Omega, \end{cases}$$ where $\Delta_Q(\cdot) := \mathrm{div}_{\mathbb{H}}\bigl(|\nabla_{\mathbb{H}}(\cdot)|^{Q-2}\nabla_{\mathbb{H}}(\cdot)\bigr)$ denotes the $Q$-Laplacian on the Heisenberg group $\mathbb{H}^N$, $\Omega \subset \mathbb{H}^N$ is a smooth bounded domain with boundary $\partial\Omega$, $f$ behaves like exponential growth in the sense of Moser-Trudinger, and $\alpha > \frac{1}{2}$. The objectives of the paper are twofold: first, to establish the existence of a nontrivial positive weak solution, and subsequently to obtain least energy nodal (sign-changing) solutions under both subcritical and critical exponential growth assumptions on the nonlinearity $f$. The analysis relies on a suitable change of variables that reduces the original quasilinear structure to a semilinear variational framework, together with critical point theory on appropriately defined Nehari-type manifolds. The results derived here appear to be genuinely new even in the classical Euclidean setting, thereby extending the existing theory for quasilinear Schr\"odinger-type equations to the sub-Riemannian context of the Heisenberg group.