A matching construction of self-similar profiles for the fast diffusion equation

Let $n\ge 3$, $00$, $η_0>0$, $ρ_1>0$, $β_-(ρ_1)=-\frac{ρ_1}{2}$, $β_+(ρ_1)=\frac{mρ_1}{n-2-nm}$ and $α=\frac{2β+ρ_1}{1-m}$. For any $β_-(ρ_1)\leβ\leβ_+(ρ_1)$, we construct the unique maximal positive radial branch of \[ Δ(f^m/m)+αf+βx\cdot\nabla f=0 \] issuing from prescrib