$\mathbf R^n$中Hardy-Hénon方程解的存在性和不存在性
Exhaustive existence and non-existence results for Hardy-Hénon equations in $\mathbf R^n$
摘要：
本文讨论了Hardy-Hénon方程\[ -Δu = |x|^σu^p \]在$\mathbf R ^n$中用$n \geq 1$和任意$p, σ\in \mathbf R$的解.这个方程是由Hénon在1973年提出的，作为天体物理学中研究旋转恒星系统的模型。虽然已经有许多工作致力于研究上述方程，但通常假设以下三个假设$p>1$、$σ\geq -2$和$n \geq 3$中的至少一个。本文的目的是调查这些参数的其他情况下的方程，导致一个完整的图片的存在/不存在的结果，非平凡的，非负解的参数的全部一般性。除了解的存在/不存在性结果外，还讨论了解的唯一性。
英文摘要：
This paper concerns solutions to the Hardy-Hénon equation \[ -Δu = |x|^σu^p \] in $\mathbf R ^n$ with $n \geq 1$ and arbitrary $p, σ\in \mathbf R$. 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 $p>1$, $σ\geq -2$, 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.
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