Existence and uniqueness of the singular self-similar solutions of the fast diffusion equation and logarithmic diffusion equation

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, ${\rho }_1>0$, $\eta >0$, $\beta >\frac{m{\rho }_1}{n-2-nm}$, $\alpha ={\alpha }_m=\frac{2\beta +{\rho }_1}{1-m}$, ${\beta }_0>0$ and ${\alpha }_0=2{\beta }_0+1$. We use fixed point argument to give a new proof for the existence and uniqueness of radially symmetric singular solution $f=f^{(m)}$ of the elliptic equation $\mathrm{\Delta }(f^m\mathrm{/}m)+\alpha f+\beta x\cdot \mathrm{\nabla }f=0$, $f>0$, in ${\mathbb{R}}^n\setminus \{0\}$, satisfying ${\displaystyle \underset{\mathrm{\mid }x\mathrm{\mid }\to 0}{\lim }\mathrm{\mid }x{\mathrm{\mid }}^{\alpha \mathrm{/}\beta }f(x)=\eta }$. We also prove the existence and uniqueness of radially symmetric singular solution $g$ of the equation $\mathrm{\Delta }\log g+{\alpha }_{0}g+{\beta }_{0}x\cdot \mathrm{\nabla }g=0$, $g>0$, in ${\mathbb{R}}^n\setminus \{0\}$, satisfying ${\displaystyle \underset{\mathrm{\mid }x\mathrm{\mid }\to 0}{\lim }\mathrm{\mid }x{\mathrm{\mid }}^{{\alpha }_0\mathrm{/}{\beta }_0}g(x)=\eta }$. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation $u_t=\mathrm{\Delta }{u}^m$ and the logarithmic diffusion equation $u_t=\mathrm{\Delta }\log u$ respectively. We will also prove the asymptotic decay rate of the function $f$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$.