论文标题
$ \ mathbf r^n $中Hardy-Hénon方程的详尽存在和不存在的结果
Exhaustive existence and non-existence results for Hardy-Hénon equations in $\mathbf R^n$
论文作者
论文摘要
本文涉及对Hardy -Hénon方程的解决方案\ [-ΔU= | X |^σu^p \] $ \ Mathbf r^n $，带有$ n \ geq 1 $ and nutyary $ p，σ\ in \ mathbf r $。 Hénon在1973年提出了该方程，作为研究天体物理学旋转恒星系统的模型。尽管有许多专门研究上述方程式的作品，但通常假设通常假设以下三个假设中的至少一个$ p> 1 $，$ p> 1 $，$σ\ geq -2 $和$ n \ geq 3 $。本文的目的是在这些参数的其他情况下研究方程，从而完整地了解了参数的整个一般性中非平凡的，非负解决方案的存在/不存在结果。除了存在/不存在的结果外，还讨论了解决方案的独特性。
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.
