Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry

Let $n\in \mathbb{N}$ with $n?2$, $a\in (?1,0)\cup (0,1]$ and $f:(0,1)\times (0,\infty )\to \mathbb{R}$ such that for each $u\in (0,\infty )$, $r?{(1+a{r}^2)}^{(n+2)/2}f(r,{(1+a{r}^2)}^{?(n?2)/2}u):(0,1)\to \mathbb{R}$ is nonincreasing. We show that each positive solution of$\mathrm{\Delta }u+f(|x|,u)=0\text{in}B,u=0\text{on}?B$ is radially symmetric, where B is the open unit ball in ${\mathbb{R}}^N$.