Existence of new self-similar solutions of the fast diffusion equation

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\eta >0$, ${\eta }_0>0$, ${\rho }_1>0$, $-\frac{{\rho }_1}{2}<\beta <\frac{m{\rho }_1}{n-2-nm}$ and $\alpha =\frac{2\beta +{\rho }_1}{1-m}$. We will prove the existence of radially symmetric solution of the equation $\mathrm{\Delta }(f^m\mathrm{/}m)+\alpha f+\beta x\cdot \mathrm{\nabla }f=0$, $f>0$, in ${\mathbb{R}}^n$, which satisfies $f(0)={\eta }_0$, $f_r(0)=0$. When $\beta <\frac{m{\rho }_1}{n-2-nm}$ holds instead, we will also prove the existence of radially symmetric solution of the equation $\mathrm{\Delta }(f^m\mathrm{/}m)+\alpha f+\beta x\cdot \mathrm{\nabla }f=0$, $f>0$, in ${\mathbb{R}}^n\setminus \{0\}$, which satisfies ${\lim }_{x\to \mathrm{\infty }}\mathrm{\mid }x{\mathrm{\mid }}^{\frac{n-2}{m}}f(x)=\eta$. As a consequence if $f_1$, $f_2$, are the solutions of the above two problems with ${\rho }_1=1$, then the function $V_i(x,t)=(T-t{)}^{\alpha }{f}_i(T-t{)}^{\beta }x)$, $i=1,2$, are backward similar solutions of the fast diffusion equation $u_t=\mathrm{\Delta }(u^m\mathrm{/}m)$ in ${\mathbb{R}}^n\times (-\mathrm{\infty },T)$ and $({\mathbb{R}}^n\setminus \{0\})\times (-\mathrm{\infty },T)$ respectively.