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

We study the existence of standing wave solutions for the following quasilinear Schrödinger equations with critical growth on the Heisenberg group $$-{\mathrm{\Delta }}_{\mathbb{H}}u+V(\xi )u-{\mathrm{\Delta }}_{\mathbb{H}}({\mid u\mid }^{2\alpha }){\mid u\mid }^{2\alpha -2}u=\lambda {\mid u\mid }^{q-2}u+{\mid u\mid }^{p-2}u\text{ in }{\mathbb{H}}^N$$ where ${\mathbb{H}}^N$ is Heisenberg group, ${\mathrm{\Delta }}_{\mathbb{H}}$ is Kohn Laplacian operator, $4\alpha <q<p\le 2\alpha Q^{\ast }$, $Q^{\ast }=\frac{2Q}{Q-2}$ is the critical Folland–Stein exponent, $\lambda$ and $\alpha$ are positive parameters, $\alpha >\frac{1}{2}\mathrm{.}$ 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.