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Influence of an $L^p$-perturbation on Hardy-Sobolev inequality with singularity a curve

We consider a bounded domain $Ω$ of $\mathbb{R}^N$, $N\ge3$, $h$ and $b$ continuous functions on $Ω$. Let $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the perturbed Hardy-Sobolev equation: $$ -Δu+h u+bu^{1+δ}=ρ^{-σ}_Γu^{2^*_σ-1} \qquad \textrm{ in } Ω, $$ where $2^*_σ:=\frac{2(N-σ)}{N-2}$ is the critical Hardy-Sobolev exponent, $σ\in [0,2)$, $0< δ<\frac{4}{N-2}$ and $ρ_Γ$ is the distance function to $Γ$. We show that the existence of minimizers does not depend on the local geometry of $Γ$ nor on the potential $h$. For $N=3$, the existence of ground-state solution may depends on the trace of the regular part of the Green function of $-Δ+h$ and or on $b$. This is due to the perturbative term of order ${1+δ}$.