Hardy-Sobolev inequality with singularity a curve
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.
math.AP↗