  In this paper, we study the following degenerate critical elliptic equations
with anisotropic coefficients $$ -div(|x_{N}|^{2\alpha}\nabla
u)=K(x)|x_{N}|^{\alpha\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N} $$
where $x=(x_{1},...,x_{N})\in\mathbb{R}^{N},$ $N\geq 3,$ $\alpha>1/2,$ $0\leq
s\leq 2$ and $2^{*}(s)=2(N-s)/(N-2).$ Some basic properties of the degenerate
elliptic operator $-div(|x_{N}|^{2\alpha}\nabla u)$ are investigated and some
regularity, symmetry and uniqueness results for entire solutions of this
equation are obtained. We also get some variational identities for solutions of
this equation. As a consequence, we obtain some nonexistence results for
solutions of this equation.
