  Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$,
$\beta\ge\frac{m\rho_1}{n-2-nm}$ and $\alpha=\frac{2\beta+\rho_1}{1-m}$. For
any $\lambda>0$, we will prove the existence and uniqueness (for
$\beta\ge\frac{\rho_1}{n-2-nm}$) of radially symmetric singular solution
$g_{\lambda}\in C^{\infty}(R^n\setminus\{0\})$ of the elliptic equation $\Delta
v^m+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $R^n\setminus\{0\}$, satisfying
$\displaystyle\lim_{|x|\to
0}|x|^{\alpha/\beta}g_{\lambda}(x)=\lambda^{-\frac{\rho_1}{(1-m)\beta}}$. When
$\beta$ is sufficiently large, we prove the higher order asymptotic behaviour
of radially symmetric solutions of the above elliptic equation as
$|x|\to\infty$. We also obtain an inversion formula for the radially symmetric
solution of the above equation. As a consequence we will prove the extinction
behaviour of the solution $u$ of the fast diffusion equation $u_t=\Delta u^m$
in $R^n\times (0,T)$ near the extinction time $T>0$.
