Exhaustive existence and non-existence results for Hardy-Hénon equations in ${\mathbf{R}}^n$

Abstract

This paper concerns solutions to the Hardy-Hénon equation \[ -Δu = |x|^σu^p \] in ${\mathbf{R}}^n$ with $n\ge 1$ and arbitrary $p,\sigma \in \mathbf{R}$. This equation was proposed by Hénon 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 $p>1$, $\sigma \ge -2$, and $n\ge 3$ 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.

30 pages, 0 figure