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 a 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 $ \Delta (f^m/m)+\alpha f+\beta x\cdot\nabla f = 0 $, $ f>0 $, in $ \mathbb{R}^n\setminus\{0\} $, satisfying $ \lim\limits_{|x|\to 0}|x|^{\alpha/\beta}f(x) = \eta $. We also prove the existence and uniqueness of radially symmetric singular solution $ g $ of the equation $ \Delta\log g+\alpha_0 g+\beta_0x\cdot\nabla g = 0 $, $ g>0 $, in $ \mathbb{R}^n\setminus\{0\} $, satisfying $ \lim\limits_{|x|\to 0}|x|^{\alpha_0/\beta_0}g(x) = \eta $. Such equations arise from the study of backward singular self-similar solution of the fast diffusion equation $ u_t = \Delta u^m $ and the logarithmic diffusion equation $ u_t = \Delta\log u $, respectively. We will also prove the asymptotic decay rate of the function $ f $ as $ |x|\to\infty $.