Let \(n\ge 3\) , \(0<m<\frac{n-2}{n}\) , \(\gamma >0\) and \(\eta >0\) . Suppose either (i) \(\alpha \ne 0\) and \(\beta =0\) or (ii) \(\alpha \in \mathbb {R}\) and \(\beta \ne 0\) holds. We will study the elliptic equation \(\Delta (f^m/m)+\alpha f+\beta x\cdot \nabla f=0\) , \(f>0\) , in \(\mathbb {R}^n{\setminus }\{0\}\) with \(\lim \nolimits _{r\rightarrow 0}\,r^{\gamma }f(r)=\eta \) . This equation arises from the study of the singular self-similar solutions of the fast diffusion equation which blow up at the origin. We will prove that if there exists a radially symmetric singular solution of the above elliptic equation, then either \(\gamma =\frac{2}{1-m}\) and \(\alpha >\frac{2\beta }{1-m}\) or \(\gamma >\frac{2}{1-m}\) , \(\beta \ne 0\) and \(\gamma =\alpha /\beta \) . As a consequence, we obtain the non-existence of radially symmetric self-similar solution of the fast diffusion equation \(u_t=\Delta (u^m/m)\) , \(u>0\) , which blows up at the origin with rate \(|x|^{-\gamma }\) when either \(0<\gamma \ne \frac{2}{1-m}\) and \(\gamma \ne \alpha /\beta \) , \(\alpha \in \mathbb {R}\) and \(\beta \ne 0\) or \(\gamma =\frac{2}{1-m}\) and \(\left( \alpha -\frac{2\beta }{1-m}\right) \eta ^{1-m}\ne \frac{2(n-2-nm)}{(1-m)^2}\) holds.
