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

Abstract

We consider a bounded domain $\Omega$ of ${\mathbb{R}}^N$, $N\ge 3$, $h$ and $b$ continuous functions on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u\in H_0^1\left(\Omega \right)$ to the perturbed Hardy-Sobolev equation: $$-\Delta u+hu+b{u}^{1+\delta }={\rho }_{\Gamma }^{-\sigma }{u}^{2_{\sigma }^{\ast }-1}\text{ in }\Omega ,$$ where $2_{\sigma }^{\ast }:=\frac{2(N-\sigma )}{N-2}$ is the critical Hardy-Sobolev exponent, $\sigma \in [0,2)$, $0<\delta <\frac{4}{N-2}$ 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 $1+\delta$.

arXiv admin note: substantial text overlap with arXiv:1702.02202