We study the quasilinear Dirichlet boundary problem
where
$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $
Here,
We also concern the Hénon type anisotropic Liouville equation,
$ -Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N}, $
where
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