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

TL;DR

This paper investigates how an $L^p$-perturbation affects the existence of solutions to a Hardy-Sobolev equation with a singularity along a curve in a bounded domain, revealing that existence is largely independent of local geometry and potential, with some dependence in three dimensions.

Contribution

It demonstrates that the existence of minimizers for the perturbed Hardy-Sobolev problem is generally unaffected by local geometry or potential, except in 3D where additional factors influence solutions.

Findings

Existence of minimizers is independent of local geometry of $ ext{Gamma}$.

Potential $h$ does not affect the existence of solutions.

In 3D, solutions depend on Green function trace and $b$.

Abstract

We consider a bounded domain $\Omega$ of $\mathbb{R}^N$, $N\ge3$, $h$ and $b$ continuous functions on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the perturbed Hardy-Sobolev equation: $$ -\Delta u+h u+bu^{1+\delta}=\rho^{-\sigma}_\Gamma u^{2^*_\sigma-1} \qquad \textrm{ in } \Omega, $$ where $2^*_\sigma:=\frac{2(N-\sigma)}{N-2}$ 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 ${1+\delta}$.