Quasilinear Schrödinger Critical Problem on the Heisenberg group \(\mathbb{H}^N\)

We study the existence of standing wave solutions for the following quasilinear Schr\"odinger equations with critical growth on the Heisenberg group $$-\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \left|u\right|^{q-2}u + \left|u\right|^{p-2}u \text{ in }\mathbb{H}^N$$ where $\mathbb{H}^N$ is Heisenberg group, $\Delta_{\mathbb{H}}$ is Kohn Laplacian operator, $4\alpha <q<p \leq 2\alpha Q^{*}$, $Q^{*}= \frac{2Q}{Q-2}$ is the critical Folland--Stein exponent, $\lambda$ and $\alpha$ are positive parameters, $\alpha > \frac{1}{2}.$ By a suitable nonlinear change of variables, the quasilinear equation is transformed into a semilinear one, allowing the use of variational methods in the Folland--Stein Sobolev space $S^{1,2}(\mathbb{H}^N)$. Applying the mountain pass theorem together with a concentration--compactness argument adapted to the sub-Riemannian framework, we establish the existence of a nontrivial solution.