Consider $m\>1$, $N\ge 1$ and $\max\{ -2,-N\}\<\sigma\<0$. The Hardy-H\'enon equation with sublinear absorption\begin(equation*}- \Delta v(x) - |x|^\sigma v(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 $v\in H^1(\mathbb{ R}^N)\cap L^{ (m+1)/m}(\mathbb{ R}^N)$, 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.
Categories: math.AP