Influence of the curvature in the existence of solutions for a two Hardy-Sobolev critical exponents

TL;DR

This paper investigates how the curvature of a curve within a domain influences the existence of positive solutions to a nonlinear elliptic PDE with Hardy-Sobolev critical exponents, using variational methods.

Contribution

It establishes the existence of mountain pass solutions for the PDE, highlighting the role of the curve's local geometry and the potential function.

Findings

Existence of solutions depends on the local curvature of the curve.

Solutions are obtained via mountain pass theorem.

The potential function influences solution existence.

Abstract

For $N\geq 4$, we let $\Omega$ be a bounded domain of $\mathbb{R}^N$ and $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation \begin{equation}\label{Atusi} -\Delta u+hu=\lambda\rho^{-s_1}_\Gamma u^{2^*_{s_1}-1}+\rho^{-s_2}_\Gamma u^{2^*_{s_2}-1} \qquad \textrm{ in } \Omega \end{equation} where $h$ is a continuous function and $\rho_\Gamma$ is the distance function to $\Gamma$. We prove the existence of a mountain pass solution for this Euler-Lagrange equation depending on the local geometry of the curve and the potential $h$.