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

TL;DR

This paper proves the existence and regularity of radially symmetric, compactly supported solutions to a Hardy-Hénon equation with sublinear absorption in b^N, using advanced inequalities and establishing connections to porous medium equations.

Contribution

It introduces new existence and regularity results for solutions to a Hardy-He9non equation with sublinear absorption, including extremal functions for Caffarelli-Kohn-Nirenberg inequalities.

Findings

Existence of at least one non-negative, radially symmetric, compactly supported solution.

Solutions are bounded and have improved regularity in Sobolev spaces.

Connection established to solutions of porous medium equations with singular coefficients.

Abstract

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.