Positive solutions of anisotropic Yamabe-type equations in $\mathbb {R}^n$
Positive solutions of anisotropic Yamabe-type equations in $\mathbb {R}^n$
We study entire positive solutions to the partial differential equation in $\mathbb {R}^{n}$, \[ \Delta _{x}u + (\alpha + 1)^2 |x|^{2\alpha } \Delta _{y}u = - |x|^{2\alpha } u^{\frac {n+2}{n-2}}, \] where $x\in \mathbb {R}^2$, $y\in \mathbb {R}^{n-2}$, $n\geq 3$ and $\alpha > 0$. We classify positive solutions with second …