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Existence of specific \emph{eternal solutions} in exponential self-similarform to the following quasilinear diffusion equation with strongabsorption$$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$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 symmetricsolutions 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 ofthese solutions is that they are bounded and do not vanish in finite time, aphenomenon which is known to take place for all non-negative bounded solutionswhen $\sigma\in (0,\sigma_c)$.

Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part I: Radially non-increasing profiles