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arXiv 1702.02202math.APmath.DG
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曲线具有奇异性的Hardy-Sobolev不等式

Hardy-Sobolev inequality with singularity a curve

Mouhamed Moustapha Fall, El hadji Abdoulaye Thiam

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27 pages
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摘要：

我们考虑$\mathbb{R}^N$的有界域$Ω$，$N\geq 3$，$h$是$Ω$上的连续函数.令$Γ$是包含在$Ω$中的闭合曲线。研究了方程$$ -Δu+h u=ρ^{-σ}_Γu^{2^*_σ-1} \qquad \textrm{ in } Ω$$正解$u\in H^1_0(Ω)$的存在性，其中$2^*_σ:=\frac{2(N-σ)}{N-2}$，$σ\in (0,2)$，$ρ_Γ$是到$Γ$的距离函数.对于$N\geq 4$，我们找到了一个充分条件，给出了曲线的局部几何，基态解的存在。在$N=3$情形下，只要$-Δ+h$的格林函数的正则部分的迹在曲线的某一点为正，我们就得到基态解的存在性.

英文摘要：

We consider a bounded domain $Ω$ of $\mathbb{R}^N$, $N\geq 3$, and $h$ a continuous function on $Ω$. Let $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u\in H^1_0(Ω)$ to the equation $$ -Δu+h u=ρ^{-σ}_Γu^{2^*_σ-1} \qquad \textrm{ in } Ω$$ where $2^*_σ:=\frac{2(N-σ)}{N-2}$, $σ\in (0,2)$, and $ρ_Γ$ is the distance function to $Γ$. For $N\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of the Green of $-Δ+h$ is positive at a point of the curve.

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