A Hardy-H{é}non equation in ${\mathbb{R}}^N$ with sublinear absorption

Consider $m\mathrm{\&}gt;1$, $N\ge 1$ and $\max \{-2,-N\}\text{<}\sigma \text{<}0$. The Hardy-Hénon equation with sublinear absorption\begin(equation*}- Δv(x) - |x|^σv(x) + \frac{1}{m-1} v^{1/m}(x)= 0, \qquad x\in$\mathbb{R}$^N,\end{equation*}is shown to have at least one solution $v\in H^1({\mathbb{R}}^N)\cap L^{(m+1)\mathrm{/}m}({\mathbb{R}}^N)$, which is non-negative and radially symmetric with a non-increasing profile. In addition, any such solution is compactly supported, bounded and enjoys the better regularity $v\in W^{2,q}({\mathbb{R}}^N)$ for $q\in [1,N\mathrm{/}\mathrm{\mid }\sigma \mathrm{\mid })$. A key ingredient in the proof is a particular case of the celebrated Caffarelli-Kohn-Nirenberg inequalities, for which we obtain the existence of an extremal function which is non-negative, bounded, compactly supported and radially symmetric with a non-increasing profile.A by-product of these results is the existence of compactly supported separate variables solutions to a porous medium equation with a spatially dependent source featuring a singular coefficient.