A Detailed Study of Kirchhoff-type Critical Elliptic Equations and $p$-Sub-Laplacian Operators within the Heisenberg Group ${\mathcal{H}}_n$ Framework

This article presents a comprehensive study of $\mathit{\text{Kirchhoff-type}}\mathit{\text{Critical}}\mathit{\text{Elliptic}}\mathit{\text{Equations}}$ involving $p$-sub-Laplacian Operators on the $\mathit{\text{Heisenberg}}\mathit{\text{Group}}$ ${\mathcal{H}}_n$. It delves into the mathematical framework of Heisenberg Group, and explores their Spectral Properties. A significant focus is on the existence and multiplicity of solutions under various conditions, leveraging concepts like the $\mathit{\text{Mountain}}\mathit{\text{Pass}}\mathit{\text{Theorem}}$. This work not only contributes to the theoretical understanding of such groups but also has implications in fields like Quantum Mechanics and Geometric Group Theory.