  We classify radially symmetric self-similar profiles presenting finite time
blow-up to the quasilinear diffusion equation with weighted source $$
u_t=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$,
$T>0$, in dimension $N\geq1$ and in the range of exponents $-2<\sigma<\infty$,
$1<m<p<p_s(\sigma)$, where $$
p_s(\sigma)=\left\{\begin{array}{ll}\frac{m(N+2\sigma+2)}{N-2}, & N\geq3,\\
+\infty, & N\in\{1,2\},\end{array}\right. $$ is the renowned Sobolev critical
exponent. The most interesting result is the \emph{multiplicity of two
different types} of self-similar profiles for $p$ sufficiently close to $m$ and
$\sigma$ sufficiently close to zero in dimension $N\geq2$, including
\emph{dead-core profiles}. For $\sigma=0$, this answers in dimension $N\geq2$ a
question still left open in \cite[Section IV.1.4, pp. 195-196]{S4}, where only
multiplicity in dimension $N=1$ had been established. Besides this result, we
also prove that, for any $\sigma\in(-2,0)$, $N\geq1$ and $m<p<p_s(\sigma)$
\emph{existence} of at least a self-similar blow-up profile is granted. In
strong contrast with the previous results, given any $N\geq1$,
$\sigma\geq\sigma^*=(mN+2)/(m-1)$ and $p\in(m,p_s(\sigma))$,
\emph{non-existence} of any radially symmetric self-similar profile is proved.
