Exhaustive existence and non-existence results for Hardy-H\'enon equations in $\mathbf R^n$

Abstract

This paper concerns solutions to the Hardy-H\'enon equation \[ -\Delta u = |x|^\sigma u^p \] in $\mathbf R ^n$ with $n \geq 1$ and arbitrary $p, \sigma \in \mathbf R$. This equation was proposed by H\'enon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above equation, at least one of the following three assumptions $p>1$, $\sigma \geq -2$, and $n \geq 3$ is often assumed. The aim of this paper is to investigate the equation in other cases of these parameters, leading to a complete picture of the existence/non-existence results for non-trivial, non-negative solutions in the full generality of the parameters. In addition to the existence/non-existence results, the uniqueness of solutions is also discussed.