Consider \(m>1\) , \(N\ge 1\) and \(\max \{-2,-N\}<\sigma <0\) . The Hardy-Hénon equation with sublinear absorption \(\begin{aligned} - \Delta v(x) - |x|^\sigma v(x) + \frac{1}{m-1} v^{1/m}(x)= 0, \qquad x\in {\mathbb {R}}^N, \end{aligned}\) is shown to have at least one solution \(v\in H^1({\mathbb {R}}^N)\cap L^{(m+1)/m}({\mathbb {R}}^N)\) , 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 \(q\in [1,N/|\sigma |)\) . 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.
