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

Abstract

We study the existence of standing wave solutions for the following quasilinear Schrödinger 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, $4\alpha <q<p\le 2\alpha Q^{\ast }$, \(Q^{*}= \frac{2Q}{Q-2}\) 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.

This paper has been withdrawn by the authors due to a crucial error in the proof of regularity results