Geometric Properties of a General Kohn-Nirenberg Domain in $\mathbb{C}^{n}$

The Kohn-Nirenberg domains are unbounded domains in $\mathbb{C}^{n}$. In this article, we modify the Kohn-Nirenberg domain $\Omega_{K,L} =\left\{(z_{1},\ldots, z_{n}) \right.\in \mathbb{C}^{n} : Re z_{n} + g \mid z_{n}\mid^{2} + \sum_{j = 1}^{n-1} (\mid z_{j}\mid^{p} +K_{j} \mid z_{j}\mid^{p-q} Re z_{j}^{q} + L_{j} \mid z_{j}\mid^{p-2q} Im z_{j}^{2q}) < 0 \}$ and discuss the existence of supporting surface and peak function at the origin.