Influence of an Lp-perturbation on Hardy-Sobolev inequality with singularity a curve

Abstract

We consider a bounded domain $\Omega$ of $\mathbb{R}^{N}$, $N \geq 3$, $h$ and $b$ continuous functions on $\Omega.$ Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^{1}_{0}(\Omega)$ to the perturbed Hardy-Sobolev equation: $-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,$ where $2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}$ is the critical Hardy-Sobolev exponent, $\sigma \in [0,2)$, $0\lt\delta\lt\frac{4}{N-2}$ and $\rho_{\Gamma}$ is the distance function to $\Gamma$. We show that the existence of minimizers does not depend on the local geometry of $\Gamma$ 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 $-\Delta+h$ and or on $b$. This is due to the perturbative term of order $1+\delta$.