Positive solutions of anisotropic Yamabe-type equations in $\mathbb{R}^n$

We study entire positive solutions to the partial differential equation in $\mathbb{R}^{n}$,

$$\Delta_{x}u + (\alpha + 1)^2 \vert x\vert^{2\alpha} \Delta_{y}u = - \vert x\vert^{2\alpha} u^{\frac{n+2}{n-2}},$$

where $x\in\mathbb{R}^2$, $y\in\mathbb{R}^{n-2}$, $n\geq 3$ and $\alpha > 0$. We classify positive solutions with second order derivatives satisfying a suitable growth near the set $x=0$.