论文标题
重新审视高阶Hardy-Hénon方程的更高阶段的存在和不存在的结果
Existence and non-existence results for the higher order Hardy-Hénon equation revisited
论文作者
论文摘要
本文致力于在$ \ \ m r^n $带有$ p> 1 $的$ \ mathbf r^n $中，研究了高阶Hardy-Hénon方程\ [（ - δ）^m u = | x |^σu^p \]的高阶hardy-hénon方程的研究。我们表明条件\ [n -2m- \ frac {2m+σ} {p -1}> 0 \]对于存在分布解决方案是必不可少的。对于$ n \ geq 200万$和$σ> -2M $，我们证明任何分销解决方案都满足积分方程和弱超级多谐属性。我们为穿刺或经典解决方案建立了一些足够的条件，使其成为分布解决方案。作为应用程序，我们表明，如果$ n \ geq 2m $和$σ> -2M $，则没有非负，非平凡的，经典的解决方案，如果\ [1 <p <\ frac {n+2m+2m+2σ}} {n-2m}。 \最终，我们证明，对于$ n> 2m $，$σ> -2M $和$$ p \ geq \ frac {n+2m+2σ} {n -2m}，$$存在正面的，辐射的对称性，经典的对称解决方案。
This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions to the higher order Hardy-Hénon equations \[ (-Δ)^m u = |x|^σu^p \] in $\mathbf R^n$ with $p > 1$. We show that the condition \[ n - 2m - \frac{2m+σ}{p-1} >0 \] is necessary for the existence of distributional solutions. For $n \geq 2m$ and $σ> -2m$, we prove that any distributional solution satisfies an integral equation and a weak super polyharmonic property. We establish some sufficient conditions for punctured or classical solution to be a distributional solution. As application, we show that if $n \geq 2m$ and $σ> -2m$, there is no non-negative, non-trivial, classical solution to the equation if \[ 1 < p < \frac{n+2m+2σ}{n-2m}. \] At last, we prove that for for $n > 2m$, $σ> -2m$ and $$p \geq \frac{n+2m+2σ}{n-2m},$$ there exist positive, radially symmetric, classical solutions to the equation.
