  Existence of specific \emph{eternal solutions} in exponential self-similar
form to the following quasilinear diffusion equation with strong
absorption$$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$posed for
$(t,x)\in(0,\infty)\times\mathbb{R}^N$, with $m>1$, $q\in(0,1)$ and
$\sigma=\sigma_c:=2(1-q)/(m-1)$ is proved. Looking for radially symmetric
solutions of the form$$u(t,x)=e^{-\alpha t}f(|x|e^{\beta t}), \qquad
\alpha=\frac{2}{m-1}\beta,$$we show that there exists a unique exponent
$\beta^*\in(0,\infty)$ for which there exists a one-parameter family
$(u_A)_{A>0}$ of solutions with compactly supported and non-increasing profiles
$(f_A)_{A>0}$ satisfying $f_A(0)=A$ and $f_A'(0)=0$. An important feature of
these solutions is that they are bounded and do not vanish in finite time, a
phenomenon which is known to take place for all non-negative bounded solutions
when $\sigma\in (0,\sigma_c)$.
