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: $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{Q}u-{\mathrm{\Delta }}_Q\left(\mathrm{\mid }u{\mathrm{\mid }}^{2\alpha }\right)\mathrm{\mid }u{\mathrm{\mid }}^{2\alpha -2}u=\lambda f(\xi ,u)& \text{in }\mathrm{\Omega },\\ u=0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where ${\mathrm{\Delta }}_Q(\cdot ):={\mathrm{d}\mathrm{i}\mathrm{v}}_{\mathbb{H}}(\mathrm{\mid }{\mathrm{\nabla }}_{\mathbb{H}}(\cdot ){\mathrm{\mid }}^{Q-2}{\mathrm{\nabla }}_{\mathbb{H}}(\cdot ))$ denotes the $Q$-Laplacian on the Heisenberg group ${\mathbb{H}}^N$, $\mathrm{\Omega }\subset {\mathbb{H}}^N$ is a smooth bounded domain with boundary $\mathrm{\partial }\mathrm{\Omega }$, $f$ 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 Schrödinger-type equations to the sub-Riemannian context of the Heisenberg group.