Existence and regularity result for a Heisenberg \(\varphi \)-laplacian problem without space reflexivity

This paper aims to investigate the existence of at least a weak solution for a nonlinear problem governed by Heisenberg \(\varphi \) $\phi$ -laplacian and under Dirichlet boundary conditions. We study this matter in both reflexive and non-reflexive cases of the associated Heisenberg Orlicz-Sobolev space. Furthermore, the objective is to acquire certain regularity and uniqueness outcomes based on supplementary assumptions. To accomplish this, variational and topological methods are employed. Moreover, an investigation into an eigenvalue problem is conducted utilizing Lagrange multiplier techniques.